Shmei¸seic gia to m jhma JEWRIAS ARIJMWN (D. Derizi¸thc) · 1.1.5 Orismìc. Oi akèraioi … kai...

Shmei¸seic gia to m�jhmaJEWRIAS ARIJMWN

(D. Derizi¸thc)

Perieqìmena

1 Diairetìthta 11.1 Diairetìthta . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Mègistoc Koinìc Diairèthc . . . . . . . . . . . . . . . . . . . . 9EukleÐdioc Algìrijmoc . . . . . . . . . . . . . . . . . . . . . . . 19Pr¸toi ArijmoÐ � Jemelei¸dec Je¸rhma thc Arijmhtik c . . . . 23To Pl joc kai h Katanom twn Pr¸twn . . . . . . . . . . . . . 30Grammikèc Diofantikèc Exis¸seic . . . . . . . . . . . . . . . . . 39H Katanom twn Pr¸twn . . . . . . . . . . . . . . . . . . . . . . 54Pujagìreiec Tri�dec kai h Mèjodoc Kajìdou tou Fermat . . . 66

2 Arijmhtik UpoloÐpwn 832.1 IsotimÐec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.2 Prìsjesh kai Pollaplasiasmìc stic IsotimÐec . . . . . . . . . . 882.3 Nìmoc Apaloif c kai Antistrèyimec Kl�seic . . . . . . . . . . . 992.4 Grammikèc IsotimÐec . . . . . . . . . . . . . . . . . . . . . . . . . 119

iii

Kef�laio 1

Diairetìthta

1.1 Diairetìthta

H basikìterh ènnoia sthn opoÐa basÐzontai ìla ìsa akoloujoÔn eÐnai h ex c:

1.1.1 Orismìc. 'Estw α, β ∈ Z. Ja lème ìti o β diaireÐ ton α ( ìti o α

diaireÐtai dia tou β) an up�rqei ènac akèraioc γ ètsi ¸ste

α = βγ

kai gr�foume “β|α”. S� aut thn perÐptwsh epÐshc lème ìti o α eÐnai èna pol-lapl�sio tou β kai ìti o β eÐnai ènac diairèthc tou α.

'Otan o β den diaireÐ ton α gr�foume “β - α”.

Parat rhsh. Ston prohgoÔmeno orismì, an o β eÐnai 6= 0 tìte o γ orÐzetaimonadik� (giatÐ;)

1.1.2 Prìtash. (Basikèc idiìthtec)

i) 'Estw α ∈ Z me α 6= 0. Tìte α| ± α

ii) O akèraioc 0 diaireÐtai dia k�je akèraio.

iii) O akèraioc 0 diaireÐ mìnon ton eautì tou.

1

2 Kef�laio 1. Diairetìthta

iv) O akèraioc 1 diaireÐ k�je akèraio.

v) An α, β ∈ Z kai β|α, tìte β| − α, −β|α, −β| − α kai |β|∣∣|α|.

vi) An α, β, γ ∈ Z kai β|α, α|γ, tìte β|γ.

vii) An α, β, γ ∈ Z me γ 6= 0 kai βγ|αγ, tìte β|α.

viii) An α, β ∈ Z kai β|α, tìte β|αγ, gia k�je γ ∈ Z.

ix) An α, β, γ, δ ∈ Z kai β|α, δ|γ, tìte βδ|αγ.

x) An α, β ∈ Z kai β|α, tìte βγ|αγ, gia k�je γ ∈ Z.

xi) An α, β, γ ∈ Z kai β|α, β|γ, tìte β|ακ + γλ, gia k�je κ, λ ∈ Z.

xii) An β|α kai α 6= 0, tìte |β| ≤ |α| (kai sunep¸c an β|α kai α|β tìte|α| = |β|).

Apìdeixh.

i) IsqÔei α = α · 1 kai −α = α(−1).

ii) Pr�gmati, isqÔei 0 = α · 0, gia k�je α ∈ Z.

iii) H sqèsh α = 0 · γ, dÐnei α = 0.

iv) An α = βγ, tìte −α = β(−γ), α = (−β)(−γ), −α = −β · γ kai sunep¸c|α| = |βγ| = |β||γ|.

vi) 'Eqoume γ = αγ′ kai α = βγ′′, opìte γ = βγ′′γ′.

vii) An αγ = βγγ′, γ′ ∈ Z, autì shmaÐnei ìti β|α.

viii) An α = βγ′, tìte kai αγ = βγ′γ.

ix) An α = βγ′ kai γ = δγ′′, tìte αγ = βδγ′γ′′.

x) An α = βγ′, tìte αγ = βγγ′.

1.1. Diairetìthta 3

ix) An α = βγ′ kai γ = βγ′′, tìte ακ = βγ′κ kai γλ = βγ′′λ, opìte ακ+γλ =

β(γ′κ + γ′′λ).

xii) 'Estw ìti α = βγ me α 6= 0, (opìte o γ prèpei na eÐnai 6= 0). Epeid o α

eÐnai akèraioc ja prèpei |γ| ≥ 1, opìte èqoume |α| = |β||γ| ≥ |β|.

Parat rhsh. H idiìthta v) anafèrei ìti “β|α an kai mìnon an |β|∣∣|α|”. Su-nep¸c h diairetìthta stouc akèraiouc an�getai sth diairetìthta stouc mh ar-nhtikoÔc akeraÐouc. Gi� autì to lìgo suqn�, ìtan melet�me probl mata diaire-tìthtac, mporoÔme na periorizìmeja stouc mh arnhtikoÔc akeraÐouc.

ParadeÐgmata.

1. Na brejoÔn ìloi oi jetikoÐ akèraioi n gia touc opoÐouc isqÔei

n + 1|n2 + 1.

Epeid n2 + 1 = (n + 1)(n − 1) + 2, gia na isqÔei n + 1|n2 + 1 prèpein + 1|2. Sunep¸c prèpei n + 1 = 1 n + 1 = 2 kai epeid o n prèpei naeÐnai ≥ 1, h lÔsh eÐnai n = 1.

2. An o n eÐnai �rtioc tìte o 4 diaireÐ ton n2 + 2n + 4. Pr�gmati, to 2|n kai2|n �ra o 2 ·2 = 4|n2. EpÐshc isqÔei 2|n kai 2|2 �ra 2 ·2 = 4|2n. Sunep¸c4|n2 + 2n kai �ra 4|n2 + 2n + 4.

3. To ginìmeno n diadoqik¸n akeraÐwn diaireÐtai dia tou n!. Upojètoume ìtiìloi oi diadoqikoÐ akèraioi m + 1, m + 2, . . . ,m + n eÐnai jetikoÐ. Opìteèqoume (

m + n

n

)=

(m + n)(m + n− 1) · · · (m + 1)n!

∈ Z.

An oi κ ap� autoÔc den eÐnai jetikoÐ, tìte to prohgoÔmeno ginìmeno topollaplasi�zoume me (−1)κ gia na gÐnei jetikì pou diaireÐtai dia n!, opìtekai to arqikì diaireÐtai dia n!. Gia par�deigma, to 3! = 6 diaireÐ k�jearijmì thc morf c n(n− 1)(n+ 1) = n3−n, en¸ to 125 = 5! diaireÐ k�jearijmì thc morf c n5 − 5n3 + 4n (giatÐ;).


To epìmeno je¸rhma apoteleÐ th jemelei¸dh idiìthta thc diairetìthtac p�nwsthn opoÐa sthrÐzetai ìlh h an�ptuxh thc stoiqei¸douc jewrÐac arijm¸n.

1.1.3 Je¸rhma (EukleÐdhc). Gia α, β ∈ Z me β 6= 0, up�rqoun monadikoÐakèraioi π kai v me 0 ≤ v < |β| tètoioi ¸ste

α = πβ + v.

Apìdeixh. JewroÔme to sÔnolo Y = {α + κβ/κ ∈ Z}. To Y perièqei mh a-rnhtikoÔc akeraÐouc, gia par�deigma o akèraioc α + (|α| + 1)|β| eÐnai ènac ap�autoÔc, (giatÐ;).

Apì thn Arq tou ElaqÐstou, to Y perièqei ènan el�qisto mh arnhtikìarijmì v. 'Estw v = α + κβ, opìte α = πβ + v, ìpou π = −κ ∈ Z. EpÐshcisqÔei v < |β| diìti diaforetik� o akèraioc v − |β| ja tan ènac mh arnhtikìcakèraioc pou an kei sto Y kai eÐnai mikrìteroc apì ton v. Sunep¸c 0 ≤ v < |β|.

T¸ra gia thn monadikìthta twn π kai v, èstw ìti èqoume kai α = βπ′ + v′

me π′, v′ ∈ Z kai 0 ≤ v′ < |β|. Opìte β(π′ − π) = v − v′. An tan v 6= v′, èstwv > v′, tìte, apì thn 1.1.2 xii), èqoume ìti |β| ≤ v− v′. All� epeid v, v′ < |β|eÐnai kai v − v′ < |β|. Sunep¸c den mporeÐ na isqÔei v 6= v′. 'Ara v = v′, opìtekai π = π′, afoÔ β 6= 0.

1.1.4 Pìrisma. 'Estw α, β ∈ Z me β 6= 0. Tìte up�rqoun monadikoÐ akèraioiπ, v me

−12|β| < v ≤ 1

2|β|

tètoioi ¸ste α = πβ + v.

Apìdeixh. Apì to 1.1.3 up�rqoun monadikoÐ akèraioi π1, v1 me 0 ≤ v1 < |β|tètoioi ¸ste

α = π1β + v1.

An v1 ≤ 12|β|, tìte jètoume v1 = v kai π1 = π. An

12|β| < v1 < |β|, tìte jètou-

me v = v1 − |β| kai π =

{π1 + 1 an |β| = β

π1 − 1 an |β| = −β. H monadikìthta prokÔptei

ìpwc kai sto 1.1.3.


1.1.5 Orismìc. Oi akèraioi π kai v sto 1.1.3 onom�zontai antÐstoiqa tophlÐko kai to upìloipo thc EukleÐdeiac diaÐreshc ( apl� thc diaÐreshc) tou α

dia tou β.

Parathr seic.

1. An kai to Je¸rhma 1.1.3 eÐnai èna je¸rhma “Ôparxhc kai monadikìthtac”pollèc forèc anafèretai wc o “Algìrijmoc DiaÐreshc” (Algìrijmoc eÐnaimia mèjodoc sthn opoÐa epanalamb�netai suneq¸c mia basik diadikasÐ-a gia th lÔsh enìc probl matoc). Ed¸ aut h onomasÐa dikaiologeÐtaiapì thn apìdeixh tou 1.1.3 kaj¸c mporoÔme na kajorÐsoume to upìloipoxekin¸ntac apì mia mh arnhtik diafor� α−κβ ≥ 0 kai diadoqik� na afai-roÔme pollapl�sia tou β tìsec forèc ìsec apaitoÔntai gia na fj�soumese mia tètoia diafor� pou na eÐnai mikrìterh tou |β|, dhlad sto upìloipov = α− πβ.

2. UpenjumÐzoume ìti wc majhtèc sto sqoleÐo gia na diairèsoume ènan akè-raio α dia enìc jetikoÔ akèraiou β efarmìzoume thn ex c algorijmhti-k mèjodo: 'Estw ìti o α èqei n + 1 yhfÐa αn, αn−1, . . . , α0, dhlad α = αnαn−1 · · ·α0 eÐnai h dekadik par�stash tou α. Sth diaÐresh tousqoleÐou ekteloÔme n+1 b mata kaj¸c k�je yhfÐo tou α apaiteÐ akrib¸cèna b ma pou dÐnei èna yhfÐo tou phlÐkou. Akribèstera, to b ma i eÐnaito ex c: BrÐskoume to megalÔtero akèraio πi tètoion ¸ste o βπi na mhneÐnai megalÔteroc tou Ai ìpou o Ai orÐzetai wc ex c:

An = αn kai Ai = 10(Ai+1 − βπi+1) + αi, 0 ≤ i < n.

Gr�foume πi sta dexi� tou πi+1. Met� to b ma i = 0 autì pou mènei eÐnaito upìloipo v. Sth diadikasÐa aut o diairejèntac α prèpei na isoÔtai meto �jroisma tou upoloÐpou kai ìlwn twn arijm¸n pou èqoun afairejeÐ.All� oi arijmoÐ pou afairoÔntai eÐnai oi βπi epÐ 10i. Sunep¸c

α = βπn10n + βπn−110n−1 + · · ·+ βπ110 + βπ0 + v

= β(πnπn−1 · · ·π0) + v = βπ + v, ìpou 0 ≤ v < β


'Etsi blèpoume ìti h diaÐresh tou sqoleÐou mac dÐnei to swstì phlÐko kaiupìloipo thc EukleÐdeiac diaÐreshc tou Jewr matoc 1.1.3.

Prèpei ed¸ na tonÐsoume ìti sto sqoleÐo de gr�foume th diaÐresh tou α

dia tou β wc α = βπ + v all� sun jwc wcα

β= π +

v

βenno¸ntac th

diaÐresh wc “pr�xh” paÐrnontac ènan rhtì arijmì. Ed¸ ìmwc lègontacìti diairoÔme ton α dia tou β den ennooÔme thn pr�xh thc diaÐreshc all�jewroÔme akrib¸c autì pou anafèretai sto Je¸rhma 1.1.3, kaj¸c stoucakèraiouc arijmoÔc den orÐzetai h pr�xh thc diaÐreshc.

3. Upojètoume ìti α > 0 kai β > 0. Tìte to pl joc twn jetik¸n polla-plasÐwn tou β pou eÐnai mikrìtera apì ton α eÐnai akrib¸c π. Pr�gmati,èna jetikì pollapl�sio λβ eÐnai mikrìtero apì ton α an kai mìnon an0 < λ ≤ α

β. All�

α

β= π +

v

β. 'Ara to pl joc twn fusik¸n arijm¸n pou

eÐnai mikrìteroi apì tonα

βeÐnai π, afoÔ 0 ≤ v < β, dhlad 0 ≤ v

β< 1.

EpÐshc, autì mac lèei ìti o π eÐnai o mikrìteroc fusikìc arijmìc pou eÐnaimikrìteroc tou

α

β, dhlad ìpwc lème, eÐnai to akèraio mèroc tou kl�smatoc

α

βkai to sumbolÐzoume me

[α

β

]. Gia to akèraio mèroc enìc rhtoÔ arijmoÔ

ja anaferjoÔme sta epìmena.

ParadeÐgmata.

1. Na brejeÐ to phlÐko kai to upìloipo thc diaÐreshc tou 59 dia tou 7. 'Eqou-me 59 = 8 ·7+3, To upìloipo eÐnai 3, 0 ≤ 3 < 7 kai to phlÐko eÐnai 8. SthdiaÐresh tou −59 dia tou 7, epeid −59 = (−9) · 7 + 4, to upìloipo eÐnaito 4 kai to phlÐko eÐnai to −9. An diairèsoume to 59 dia tou −7, tìte toupìloipo eÐnai 3 kai to phlÐko eÐnai −8 kai an diairèsoume to −59 dia tou−7, tìte to upìloipo eÐnai 4 kai to phlÐko eÐnai 9.

K�nontac tic prohgoÔmenec diarèseic sÔmfwna me to Pìrisma 1.1.4, èqoume

59 = 8 · 7 + 3, 3 <127.

−59 = (−8) · 7− 3, −3 = 4− 7 > −127 kai − 8 = −9 + 1 afoÔ 7 > 0.


59 = (−8) · 7 + 3, 3 <127.

−59 = 8(−7)− 3, −3 = 4− 7 > −127 kai 8 = 9− 1 afoÔ − 7 < 0.

2. SÔmfwna me th diaÐresh tou EukleÐdh 1.1.3, an diairèsoume ènan akèraioα dia tou 4, paÐrnoume tèssera dunat� upìloipa: to 0, to 1, to 2 kai to3. Autì shmaÐnei ìti o α mporeÐ na grafteÐ se mia apì tic ex c morfèc:

α = 4π + 0, α = 4π + 1, α = 4π + 2, α = 4π + 3.

Genik�, an diairèsoume to α dia enìc akeraÐou n tìte to upìloipo ja eÐnaiènac apì touc n arijmoÔc: 0, 1, 2, . . . , n − 1. Autì shmaÐnei ìti h diaÐre-sh tou EukleÐdh taxinomeÐ ìlouc touc akèraiouc s� autoÔc pou af nounupìloipo 0, s� autoÔc pou af noun upìloipo 1, . . . , kai s� autoÔc pouaf noun upìloipo n − 1. Gia par�deigma, an to n = 2, tìte oi akèraioitaxinomoÔntai stouc �rtiouc, dhlad s� autoÔc pou af noun upìloipo 0,kai stouc perittoÔc dhlad s� autoÔc pou af noun upìloipo 1. 'Ara ènacakèraioc eÐnai �rtioc perittìc. PÐsw ap� aut thn idèa thc taxinìmhshctwn akeraÐwn, brÐsketai h arijmhtik twn isotimi¸n pou anaptÔqjhke apìton Gauss kai ja melet soume sto epìmeno kef�laio.

3. Ac jewr soume ènan perittì arijmì α = 2κ + 1. An κ = 2κ1, tìteα = 4κ1 + 1 kai an κ = 2κ1 + 1, tìte α = 4κ1 + 3 = 4(κ1 + 1) − 1.'Ara α2 = 8(2κ ± 1)κ + 1. Dhlad to tetr�gwno enìc akèraiou perittoÔarijmoÔ eÐnai p�nta thc morf c 8π + 1 gia k�poio π ∈ Z.Sunep¸c an α kai β eÐnai dÔo perittoÐ akèraioi, tìte h diafor� α2 − β2

diaireÐtai p�nta dia tou 8.

4. Ac deÐxoume ìti metaxÔ tri¸n akèraiwn arijm¸n p�nta mporoÔme na dialè-xoume dÔo apì autoÔc ètsi ¸ste h diafor� α3β−αβ3 na diaireÐtai dia tou10. Pr�gmati èqoume

α3β − αβ3 = αβ(α− β)(α + β)


pou eÐnai ènac �rtioc arijmìc. An ènac apì touc treic arijmoÔc diaireÐtaidia 5, dhlad eÐnai thc morf c 5κ, tìte o (5κ)3β − 5κβ3 = 5(κβ(5κ −β)(5κ + β)) diaireÐtai dia tou 10 afoÔ o κβ(5κ − β)(5κ + β) eÐnai p�nta�rtioc. An kanènac apì touc treic arijmoÔc den diaireÐtai dia tou 5 tìteautìc eÐnai thc morf c 5κ ± 1 5κ ± 2 kai sunep¸c oi dÔo apì autoÔcanagkastik� ja prèpei na eÐnai thc morf c 5κ ± 1 5κ ± 2. Sunep¸c hdiafor� touc to �jroism� touc ja prèpei na eÐnai pollapl�sio tou 5.

5. To teleutaÐo yhfÐo tou tetrag¸nou enìc akèraiou arijmoÔ eÐnai ènac apìtouc arijmoÔc 0, 1, 4, 5, 6 9. Pr�gmati, k�je akèraioc α gr�fetai wcα = 10π + v, 0 ≤ v ≤ 9, ìpou to v eÐnai to teleutaÐo yhfÐo tou. 'Eqoumeα2 = 100π2 + 20πv + v2. Sunep¸c to teleutaÐo yhfÐo tou α2 eÐnai toÐdio me ekeÐno tou v2. All� ta tetr�gwna twn arijm¸n 0, 1, 2, . . . , 9 èqounteleutaÐo yhfÐo ènan apì touc arijmoÔc 0,1,4,5,6 9.

6. UpenjumÐzoume ìti ènac pragmatikìc arijmìc r eÐnai rhtìc an kai mìnonan r =

α

β, α, β ∈ Z, β 6= 0. 'Enac pragmatikìc arijmìc pou den eÐnai rhtìc

lègetai �rrhtoc. EÐnai gnwstì ìti o pr¸toc pou apèdeixe thn Ôparxhtwn �rrhtwn arijm¸n tan o Pujagìrac ( h Sqol tou) o opoÐoc mèswtou Pujagìriou Jewr matoc apèdeixe ìti o

√2 den eÐnai rhtìc. Ed¸

dÐnoume wc pr¸th apìdeixh (�llec dÐdontai sta epìmena) tou gegonìtocautoÔ, qrhsimopoi¸ntac to diaqwrismì twn fusik¸n arijm¸n se �rtiouckai perittoÔc. Upojètoume ìti

√2 =

α

β, α, β ∈ Z. ApaleÐfontac touc

koinoÔc par�gontec twn α kai β mporoÔme na jewr soume ìti o α kai oβ den èqoun koinoÔc par�gontec. Sunep¸c an o α eÐnai �rtioc ( o β

eÐnai �rtioc) tìte o β eÐnai perittìc (antÐstoiqa o α eÐnai perittìc). All�èqoume 2β2 = α2 kai sunep¸c o α2 eÐnai �rtioc. 'Ara o α eÐnai �rtioc(giatÐ;), èstw α = 2κ. Opìte β2 = 2κ2, dhlad kai o β eÐnai �rtioc poueÐnai �topo sÔmfwna me thn upìjesh.

'Alloi gnwstoÐ �rrhtoi arijmoÐ eÐnai o π(= 3, 14159 · · · ) kai o e(= 2, 71828 · · · ).Gia ton e eÐnai gnwst h ex c sÔntomh apìdeixh (oi gnwstèc apodeÐxeic


gia to π eÐnai perissìtero polÔplokec). O e orÐzetai wc

e =∞∑

n=1

1n!

.

Upojètoume ìti e =α

β, ìpou α kai β eÐnai akèraioi pou den èqoun koinoÔc

par�gontec. 'Estw

p = 1 +12!

+13!

+ · · ·+ 1β!

kai

q =1

(β + 1)!+

1(β + 2)!

+ · · ·

opìte e = p + q kai β!e = β!p + β!q. Oi arijmoÐ β!e kai β!p eÐnai akèraioikai sunep¸c kai o β!q = β!e− β!p eÐnai akèraioc. All� èqoume

0 < β!q =1

β + 1+

1(β + 1)(β + 2)

+1

(β + 1)(β + 2)(β + 3)+ · · ·

<12

+14

+18

+ · · · = 1

(gewmetrik prìodoc). Autì eÐnai �topo. 'Ara o e eÐnai �rrhtoc.

Mègistoc Koinìc Diairèthc

Apì thn idiìthta 1.1.2 xii) prokÔptei ìti to pl joc twn diairet¸n enìc 6= 0

akeraÐou arijmoÔ eÐnai peperasmèno. Sunep¸c metaxÔ ìlwn twn koin¸n diairet¸ndÔo akeraÐwn arijm¸n α kai β apì touc opoÐouc toul�qiston ènac eÐnai 6= 0,up�rqei ènac mègistoc (�ra monadikìc) pou eÐnai jetikìc (afoÔ o 1 eÐnai koinìcdiairèthc).

Par�deigma. Na brejoÔn ìloi oi jetikoÐ koinoÐ diairètec δ tou n2 + 1 kai(n+1)2+1. 'Estw δ|n2+1 kai δ|n2+2n+2, opìte δ|n2+2n+2−(n2+1) = 2n+1.'Ara δ|(2n+1)2 = 4n2 +4n+1 kai sunep¸c δ|4(n2 +2n+2)− (4n2 +4n+1) =

4n + 7. Opìte δ|4n + 7− 2(2n + 1) = 5. Dhlad o δ mporeÐ mìno na eÐnai o 1 o 5. Gia n = 1 o δ eÐnai o 1, en¸ gia n = 2 o δ eÐnai o 1 kai o 5.


1.1.6 Orismìc. 'Estw α, β ∈ Z, me toul�qiston ènan 6= 0. O mègistoc koinìcdiairèthc twn α kai β pou ja ton sumbolÐzoume m.k.d.(α, β) h apl� me (α, β),eÐnai o jetikìc akèraioc δ pou ikanopoieÐ tic dÔo idiìthtec

1. δ|α kai δ|β2. an γ|α kai γ|β tìte γ ≤ δ.

Gia par�deigma, oi arijmoÐ ±1,±2,±7,±14 eÐnai oi diairètec tou 14 kai oi±1,±5,±7,±35 eÐnai oi diairètec tou −35. Sunep¸c oi koinoÐ diairètec tou 14kai −35 eÐnai oi ±1,±7. 'Ara m.k.d.(14,−35) = 7.

T¸ra ac jewr soume ìla ta koin� pollapl�sia dÔo akèraiwn arijm¸n α kaiβ (pou eÐnai kai oi dÔo 6= 0). Tètoia up�rqoun, gia par�deigma oi arijmoÐ αβκ,κ ∈ Z. Apì thn arq tou el�qistou, up�rqei èna el�qisto koinì pollapl�siotwn α kai β sto sÔnolo twn jetik¸n koin¸n pollaplasÐwn twn α kai β.

1.1.7 Orismìc. To el�qisto koinì pollapl�sio twn α kai β, α, β ∈ Z, α 6= 0,β 6= 0, eÐnai o jetikìc akèraioc ε pou sumbolÐzetai me [α, β] kai ikanopoieÐ ticex c idiìthtec:

1. α|ε kai β|ε2. an α|m kai β|m tìte ε ≤ m.

Gia par�deigma, [5,−15] = 15, [5, 21] = 105.

ShmeÐwsh. Ap� ton orismì prokÔptei �mesa ìti (α, β)|[α, β].To epìmeno je¸rhma qarakthrÐzei ton mègisto koinì diairèth kai to el�qisto

koinì pollapl�sio dÔo akèraiwn arijm¸n.

1.1.8 Je¸rhma.

i) Ta koin� pollapl�sia dÔo akèraiwn arijm¸n α 6= 0, β 6= 0 eÐnai ta Ðdia meaut� tou [α, β].

ii) 'Enac koinìc diairèthc dÔo arijm¸n α, β ∈ Z me toul�qiston ton ènan 6= 0

eÐnai o m.k.d.(α, β) an kai mìnon an autìc diaireÐtai dia k�je koinì diairèthtwn α kai β. Dhlad oi koinoÐ diairètec twn α kai β eÐnai akrib¸c ekeÐnoitou m.k.d(α, β).


iii) IsqÔei

[α, β] =|α||β|(α, β)

.

Apìdeixh. 'Estw ε = [α, β] kai m èna koinì pollapl�sio twn α kai β. Apì to1.1.3 up�rqoun monadikoÐ π, v ∈ Z me 0 ≤ v < ε tètoia ¸ste

m = επ + v.

'Ara v = m − επ kai sunep¸c α|v kai β|v. An tan v 6= 0, tìte ja up rqe ènamikrìtero apì to ε jetikì koinì pollapl�sio twn α kai β. Autì eÐnai �topo kai�ra prèpei ε|m.

ii) An α = 0 kai β 6= 0, tìte (0, β) = |β| kai an γ|0 kai γ|β tìte γ|(0, β).An α 6= 0 kai β 6= 0, mporoÔme na upojèsoume ìti kai oi dÔo eÐnai jetikoÐ, afoÔo α èqei touc Ðdiouc diairètec me ton |α| kai to Ðdio isqÔei gia ton β. Apì thnidiìthta i) prèpei ε|αβ, èstw αβ = δε, δ ∈ Z. Ja deÐxoume ìti an γ|α kai γ|βtìte γ|δ kai ìti δ = (α, β). Profan¸c α|αβ

γkai β|βα

γ, dhlad o

αβ

γeÐnai èna

koinì pollapl�sio twn α kai β. Sunep¸c apì thn idiìthta i) èqoume ε|αβ

γ. 'Ara

oαβ

γ

/αβ

δ=

δ

γ∈ Z, dhlad γ|δ. EpÐshc

α

δ=

ε

β∈ Z kai

β

δ=

ε

α∈ Z, pou

shmaÐnei ìti δ|α kai δ|β. 'Ara o δ eÐnai o megalÔteroc koinìc diairèthc twn α kaiβ, dhlad δ = (α, β).

(Tautìqrona èqoume deÐxei kai thn iii)). To antÐstrofo eÐnai profanèc.

'Ena shmantikì apotèlesma pou qarakthrÐzei to m.k.d(α, β) kai qrhsimopoieÐ-tai suqn� gia th lÔsh problhm�twn eÐnai to ex c.

1.1.9 Je¸rhma (Bachet–Bezout, Grammik Morf tou m.k.d.). O akèraiocarijmìc δ eÐnai o m.k.d.(α, β) (ìpou α 6= 0 kai β 6= 0) an kai mìnon an o δ eÐnai omikrìteroc jetikìc akèraioc metaxÔ ìlwn twn jetik¸n arijm¸n pou mporoÔn naekfrasjoÔn sth grammik morf

αx + βy, x, y ∈ Z.

Dhlad oi arijmoÐ thc morf c αx + βy, x, y ∈ Z, eÐnai ta pollapl�sia toum.k.d(α, β).


Apìdeixh. 'Estw γ o mikrìteroc jetikìc akèraioc metaxÔ ìlwn twn jetik¸nakèraiwn thc morf c αx + βy, x, y ∈ Z. Apì thn arq tou el�qistou, tètoioc γ

up�rqei afoÔ to sÔnolo∑

= {αx + βy ∈ N/x, y ∈ Z}

eÐnai 6= ∅ (gia par�deigma o |α| ∈ ∑, an α 6= 0). 'Estw γ = αx0 + βy0.

Diair¸ntac to α dia tou γ èqoume

α = γπ + v, 0 ≤ v < γ

opìte v = α − γπ = α(1 − x0π) + β(−y0π). An v 6= 0, blèpoume ìti v ∈ ∑,

pou eÐnai �topo giatÐ v < γ. Sunep¸c prèpei v = 0. 'Ara γ|α kai to Ðdio isqÔeigia to β, dhlad γ|β. Prèpei sunep¸c γ|(α, β) = δ. Fusik� isqÔei (α, β)|γ kaitelik� (α, β) = γ.

T¸ra k�je pollapl�sio κδ eÐnai thc morf c α(κx0) + β(κy0) = αx + βy,x, y ∈ Z kai fusik� k�je akèraioc thc morf c αx + βy eÐnai pollapl�sio touδ.

Parat rhsh. An α, β, x, y ∈ Z, tìte up�rqoun �peira to pl joc zeug�ria(x′, y′), x′, y′ ∈ Z, tètoia ¸ste αx + βy = αx′ + βy′. Pr�gmati, an γ eÐnai ènackoinìc diairèthc twn α kai β (p.q. o 1) èstw α = α′γ kai β = β′γ tìte

αx + βy = α(x− β′t) + β(y + α′t) = αx′ + βy′

ìpou x′ = x− β′t kai y′ = y + α′t.

1.1.10 Orismìc. 'Estw α, β ∈ Z, (ìpou α 6= 0 β 6= 0). Tìte lème ìti autoÐeÐnai sqetik� pr¸toi metaxÔ touc an

m.k.d(α, β) = 1

(dhlad oi mìnoi koinoÐ diairètec touc eÐnai to 1 kai to �1).

Apì to prohgoÔmeno je¸rhma, an α kai β eÐnai sqetik� pr¸toi metaxÔ touc,up�rqoun x, y ∈ Z tètoioi ¸ste αx + βy = 1. Ap� thn �llh meri�, an gia dÔo


akeraÐouc α kai β up�rqoun akèraioi x, y ètsi ¸ste αx + βy = 1 tìte epeid (α, β)|1 kai (α, β) > 0 prèpei (α, β) = 1. Sunep¸c dÔo sqetik� pr¸toi arijmoÐqarakthrÐzontai wc ex c:

1.1.11 Pìrisma. DÔo akèraioi α kai β pou den eÐnai kai oi dÔo mhdèn, eÐnaisqetik� pr¸toi metaxÔ touc an kai mìnon an up�rqoun x, y ∈ Z tètoioi ¸steαx + βy = 1.

ShmeÐwsh. Autì to Pìrisma, ìpwc ja doÔme sta epìmena, eÐnai polÔ qr simosth lÔsh problhm�twn pou anafèrontai sth diairetìthta. Wc mia pr¸th efarmo-g autoÔ, deÐqnoume to gnwstì apotèlesma tou Pujagìra: o arijmìc

√2 den eÐ-

nai rhtìc. Diìti diaforetik� ja eÐqame√

2 =α

βme (α, β) = 1, opìte αx+βy = 1

gia x, y ∈ Z. 'Etsi ja eÐqame√

2 =√

2(αx + βy) =√

2αx +√

2βy = 2βx + αy,dhlad

√2 ∈ Z pou eÐnai �topo.

Wc èna epiplèon par�deigma deÐqnoume ìti

m.k.d.(n! + 1, (n + 1)! + 1) = 1

gia k�je n ∈ N. Pr�gmati, epeid èqoume

n = n + 1 + (n + 1)!− (n + 1)!− 1 = (n + 1)(n! + 1)− ((n + 1)! + 1),

an δ eÐnai o m.k.d. twn n! + 1 kai (n + 1)! + 1 gia k�poio n ∈ N, tìte δ|n. All�δ|n! + 1 kai sunep¸c o δ ja prèpei na diaireÐ kai ton m.k.d.(n, n! + 1). All�m.k.d.(n, n! + 1) = 1, afoÔ 1 = (n! + 1) · 1 + n(−(n− 1)!). Sunep¸c δ = 1.

Sthn epìmenh prìtash diatup¸nontai merikèc aplèc all� basikèc idiìthtecpou aforoÔn ton m.k.d. kai to e.k.p.

1.1.12 Prìtash. 'Estw α, β, γ ∈ Z me α2 + β2 6= 0 (dhlad toul�qistonènac apì touc α kai β eÐnai 6= 0). Tìte isqÔoun ta ex c:

i) (α, β) = (|α|, |β|) kai [α, β] = [|α|, |β|].Idiaitèrwc isqÔei α|β an kai mìnon an (α, β) = |α| an kai mìnon an [α, β] =

|β|. EpÐshc (α, β) = [α, β] an kai mìnon an |α| = |β|.


ii) Ap� to 1.1.9, up�rqoun x, y ∈ Z ètsi ¸ste αx + βy = (α, β). EpÐshcgnwrÐzoume ìti γ|(α, β) an kai mìnon an γ|α kai γ|β. Sunep¸c èqoume

α

γx +

β

γy =

(α, β)γ

.

P�li apì to 1.1.9 prokÔptei ìti(

α

γ,β

γ

) ∣∣∣∣(α, β)

γ. Epeid ìmwc

(α, β)γ

∣∣∣∣α

γkai

(α, β)γ

∣∣∣∣β

γ, apì to 1.1.8 èqoume

(α, β)γ

∣∣∣∣(

α

γ,β

γ

)kai sunep¸c

(α

γ,β

γ

)=

(α, β)γ

.

iii) Epeid (α, β)|α kai (α, β)|κβ èqoume (α, β)|α+κβ. 'Ara (α, β)|(α+κβ)

( arkeÐ na sumper�noume ìti (α, β) ≤ (α + κβ, β) ìpwc apaiteÐ o Orismìc1.1.6). EpÐshc èqoume (α + κβ, β)|κβ kai (α + κβ, β)|α + κβ − κβ = α. 'Ara(α + κβ, β)|(α, β). Sunep¸c (α, β) = (α + κβ, β).

iv) Apì to 1.1.9, up�rqoun x1, x2 ∈ Z ètsi ¸ste (γα, γβ) = γαx1 +γβx2 =

γ(αx1+βx2). P�li apì to 1.1.9, prèpei (α, β)|αx1+βx2, opìte γ(α, β)|γ(αx1+

βx2) = (γα, γβ) kai sunep¸c |γ|(α, β)|(γα, γβ). EpÐshc up�rqoun x′1, x′2 ∈ Z

ètsi ¸ste (α, β) = αx′1 + βx′2 kai �ra |γ|(α, β) = |γ|αx′1 + |γ|βx′2. Sunep¸capì to 1.1.9, (γα, γβ)||γ|(α, β). 'Ara (γα, γβ) = |γ|(α, β).

v) Apì to iv), èqoume (αγ, βγ) = |γ|. Profan¸c α|αγ kai apì thn upìjeshα|βγ. 'Ara α||γ|, opìte α|γ.Mia �llh �mesh apìdeixh eÐnai h ex c. Apì to 1.1.10 up�rqoun x, y ∈ Z ètsi¸ste αx + βy = 1 opìte γαx + γβy = γ kai epeid α|αγ kai α|βγ ja prèpeiα|γ.

vi) Epeid (α, (α, β)γ)|(α, β)|γ| = (αγ, βγ) ja prèpei (α, (α, β)γ)|βγ. Al-l� (α, (α, β)γ)|α opìte (α, (α, β)γ)|(α, βγ). 'Eqoume ìmwc (α, βγ)|α opìte(α, βγ)|αγ kai (α, βγ)|βγ. Sunep¸c (α, βγ)|(αγ, βγ) = (α, β)|γ| kai epeid (α, βγ)|α ja prèpei (α, βγ)|(α, (α, β)γ).

vii) Epeid (β, γ)|γ kai γ|α prèpei (β, γ)|α. All� (β, γ)|β. Sunep¸c (β, γ) =

1, afoÔ oi mìnoi koinoÐ diairètec tou α kai β eÐnai to ±1.viii) Pr�gmati, up�rqoun x, y ∈ Z ètsi ¸ste αx+βy = 1, opìte γαx+γβy =

γ. All� α|α kai apì thn upìjesh β|γ, �ra αβ|αγ. Gia ton Ðdio lìgo αβ|βγ.Sunep¸c αβ|αγx + βγy = γ. Autì prokÔptei epÐshc �mesa apì to 1.1.8 iii)afoÔ [α, β]|γ kai [α, β] = αβ.


ix) 'Estw (α, γ) = αx1 + βy1, (β, γ) = βx2 + γy2, x1, x2, y1, y2 ∈ Z. Opì-te (α, γ)(β, γ) = αβx1x2 + γ(αx1y2 + βy1x2 + γy1y2) kai epeid γ|αβ prèpeiγ|(α, γ)(β, γ). x) Epeid γ|α, β, apì thn ix) prèpei γ|(α, γ)(β, γ). 'Omwc (α, γ)|γkai (β, γ)|γ. ParathroÔme epÐshc ìti an ρ eÐnai ènac koinìc diairèthc twn (α, γ)

kai (β, γ), dhlad ènac diairèthc tou m.k.d((α, γ), (β, γ)), autìc prèpei na eÐnaiènac koinìc diairèthc twn α kai β. All� ap� thn upìjesh oi mìnoi koinoÐ diairètectwn α kai β eÐnai ±1. Sunep¸c ((α, γ), (β, γ)) = 1. 'Ara ap� thn viii) prokÔpteiìti ((α, γ), (β, γ))|γ opìte γ = (α, γ)(β, γ) = γ1γ2, γ1 = (α, γ), γ2 = (β, γ).DeÐqnoume t¸ra thn monadikìthta. 'Estw γ′1, γ

′2 ∈ Z me γ′1|α, γ′2|β kai γ = γ′1γ

′2.

Profan¸c γ′1|(α, γ) kai γ′2|(β, γ). An tan γ′1 6= (α, γ), tìte γ′1 < (α, γ), opìteγ = γ′1γ

′2 < (α, γ)(β, γ) = γ, �topo. To Ðdio prokÔptei fusik� kai an tan

γ′2 6= (β, γ). 'Ara prèpei γ′1 = (α, γ) kai γ′2 = (β, γ).

Shmei¸noume ìti ta prohgoÔmena apodeiknÔontai kai me �llouc trìpouc qw-rÐc th qr sh tou Jewr matoc 1.1.9 (p.q. bl. Landau [?]).

ParadeÐgmata.

1. Na deiqjeÐ ìti gia k�je n ∈ N kai α, β ∈ Z me α2 + β2 6= 0 isqÔei

(α, β)n = (αn, βn).

Kat� arq�c upojètoume ìti (α, β) = 1. Efarmìzoume epagwg sto n. Gian = 1 profan¸c isqÔei. Upojètoume ìti (αn, βn) = 1. Tìte lìgw thc1.1.12.vi) èqoume

(αn+1, βn+1) = (αnα, βnβ) = (αnα, (αnα, βn)β)

= (αnα, (α(αn, βn), βn)β) = (αnα, (α, βn)β).

All� (α, βn) = 1, kaj¸c an δ|(α, βn), tìte δ|αn kai δ|βn opìte δ|(αn, βn) =

1, dhlad δ = ±1. 'Ara

(αn+1, βn+1) = (αnα, β) = (α(αn, β), β) = (α, β) = 1

(afoÔ kai p�li (αn, β) = 1). Sunep¸c an (α, β) = 1, tìte (αn, βn) =

(α, β)n = 1, gia k�je n ∈ N.


GnwrÐzoume apì to 1.1.12.ii) ìti(

α

(α, β),

β

(α, β)

)= 1

opìte lìgw tou prohgoÔmenou apotelèsmatoc èqoume(

αn

(α, β)n,

βn

(α, β)n

)=

((α

(α, β)

)n

,

(β

(α, β)

)n)= 1.

Lìgw tou 1.1.12.iv) paÐrnoume

(α, β)n = (α, β)n

(αn

(α, β)n,

βn

(α, β)n

)= (αn, βn).

2. Na deiqjeÐ ìti gia k�je n ∈ N kai α, β ∈ Z, α > 0, β > 0, isqÔei: αn|βn

an kai mìnon an α|β.

Profan¸c an α|β tìte αn|βn. 'Estw ìti αn|βn. Tìte (α, β)n αn

(α, β)nγ =

(α, β)n βn

(α, β)ngia k�poio γ ∈ Z. Opìte

αn

(α, β)nγ =

βn

(α, β)n, �ra

αn

(α, β)n= 1, afoÔ

(αn

(α, β)n,

βn

(α, β)n

)= 1.

'Ara αn = (α, β)n, opìte α = (α, β) (giatÐ;) kai sunep¸c α|β.

3. Na deiqjeÐ ìti(

αn − βn

α− β, α− β

)= (n(α, β)n−1, α− β).

IsqÔeiαn − βn

α− β= αn−1 + αn−2β + · · ·+ αβn−2 + βn−1. (giatÐ;).

Prosjètoume kai afairoÔme apì to dexÐ mèloc to βn−1. Opìte

αn − βn

α− β= (αn−1 − βn−1) + (αn−2β − βn−1) + · · ·+ (αβn−2 − βn−1)

+ nβn−1 = (α− β)

(n−2∑

0

αn−2−iβi

)+ (α− β)β

(n−3∑

0

αn−3−iβi

)

+ (α− β)βn−2 + nβn−1 = (α− β)κ + nβn−1.


Apì to 1.1.12 iii) èqoume(αn − βn

α− β, α−β

)=

((α− β)κ + nβn−1, α− β

)

= (nβn−1, α−β). 'Omoia èqoume ìti(αn − βn

α− β, α−β

)= (nαn−1, β−α) =

(nαn−1, α−β). 'Estw d =(αn − βn

α− β, α−β

)kai d′ =

(n(α, β)n−1, α− β

).

Apì to prohgoÔmeno par�deigma èqoume ìti d′ =(n(αn−1, βn−1), α− β

)=(

(nαn−1, nβn−1), α− β). Epeid d′|(nαn−1, nβn−1), prèpei d′|nαn−1 kai

d′|nβn−1. 'Ara d′|d = (nαn−1, α − β) = (nβn−1, α − β). EpÐshc epeid d|nαn−1 kai d|nβn−1 ja prèpei d|(nαn−1, nβn−1), opìte d|d′. 'Ara d = d′.

Gia par�deigma, isqÔei

(αn − 1α− 1

, α− 1)

= (n, α− 1)

gia k�je n ∈ N. Poio genik�, an (α, β) = 1 tìte(αn − βn

α− β, α − β

)=

(n, α− β).

4. Na deiqjeÐ ìti

(nα − 1, nβ − 1) = n(α,β) − 1,

gia k�je α, β, n ∈ N me n 6= 1.

'Estw α = (α, β)γ kai β = (α, β)δ. Tìte (nα−1) = (n(α,β)−1)( γ−1∑

i=0n(α,β)i

),

dhlad n(α,β) − 1|nα − 1. 'Omoia isqÔei n(α,β) − 1|nβ − 1. Sunep¸cn(α,β) − 1|(nα − 1, nβ − 1). Apì to 1.1.9, up�rqoun x, y ∈ Z ètsi ¸ste

(α, β) = αx + βy.

Profan¸c toul�qiston ènac apì touc x kai y prèpei na eÐnai 6= 0. EpÐshcan ènac ap� touc dÔo eÐnai jetikìc, èstw x > 0, tìte y ≤ 0, diìti an tan kai y > 0, tìte ja eÐqame (α, β) = αx + βy ≥ α + β, en¸ èqoume(α, β) ≤ α kai (α, β) ≤ β. Fusik� oi x kai y den mporoÔn na eÐnai kai oidÔo arnhtikoÐ, afoÔ o (α, β) > 0. Upojètoume loipìn ìti x > 0 kai y ≤ 0.


Tìte

kai

(nα − 1, nβ − 1)|nαx − 1 = (nα − 1)x−1∑

i=0

nαi

(nα − 1, nβ − 1)|n−βy − 1 = (nβ − 1)−y−1∑

i=0

nβi.

Opìte (nα − 1, nβ − 1)|nαx − 1− n(α,β)(n−βy − 1) = n(α,β) − 1.

ShmeÐwsh. MporoÔme epÐshc na apodeÐxoume to prohgoÔmeno efarmìzontacthn EukleÐdeia diaÐresh: 'Estw α ≥ β, tìte α = βπ + v, 0 ≤ v < β. Epeid (nβπ − 1)nv = (nβ − 1)κnv, lìgw thc 1.1.12 iii) èqoume

(nα − 1, nβ − 1) = (nα − 1− (nβπ − 1)nv, nβ − 1) = (nv − 1, nβ − 1).

An suneqÐsoume thn Ðdia diadikasÐa gia ton (nv−1, nβ−1) k.o.k., ìpwc ja doÔmeamèswc met�, apì ton algìrijmo tou EukleÐdh, h diadikasÐa aut ja termatÐseiston (n(α,β) − 1, 0) = n(α,β) − 1.

EukleÐdeioc Algìrijmoc

'Estw α, β ∈ Z me α2 +β2 6= 0. Apì to Je¸rhma 1.1.9 gnwrÐzoume ìti up�rqounx, y ∈ Z tètoioi ¸ste (α, β) = αx + βy. H apìdeixh autoÔ tou jewr matoc denparèqei ènan trìpo upologismoÔ twn x kai y kai kat� epèktash tou m.k.d(α, β).Mia praktik mèjodoc �mesou upologismoÔ tou m.k.d(α, β) all� kai thc eÔreshcenìc zeugarioÔ x, y ∈ Z ètsi ¸ste (α, β) = αx + βy èdwse prin 2400 qrìnia oEukleÐdhc sto biblÐo tou “Ta StoiqeÐa tou EukleÐdh”. H mèjodoc aut sth-rÐzetai stic dÔo basikèc idiìthtec 1.1.12 i) kai iii) kai onom�zetai EukleÐdeiocAlgìrijmoc.

O algìrijmoc basÐzetai sta ex c dÔo b mata.

Upojètoume ìti α > β > 0 (kaj¸c mporoÔme). Apì thn EukleÐdeia diaÐreshup�rqoun monadikoÐ π, v ∈ Z me 0 ≤ v < β ètsi ¸ste α = βπ + v.

1o b ma: An v = 0, dhlad an β|α tìte (α, β) = β (idiìthta 1.1.12 i)).


2o b ma: An v 6= 0, tìte (α, β) = (β, v) (idiìthta 1.1.12 iii)).Sthn perÐptwsh pou eÐnai v 6= 0, epanalamb�noume thn Ðdia diadikasÐa: Efar-

mìzoume thn EukleÐdeia diaÐresh gia touc β kai v, èstw β = vπ1 + v1, ìpou0 ≤ v1 < v. An v1 = 0, tìte (α, β) = (β, v) = v, diaforetik� (α, β) = (β, v) =

(v, v1).SuneqÐzontac aut th diadikasÐa diadoqik¸n EukleÐdeiwn diairèsewn èqoume

α = βπ + v

β = vπ1 + v1

v = v1π2 + v2

v1 = v2π3 + v3

...

vi−1 = viπi+1 + vi+1

ìpou α > β > v > v1 > · · · > vi+1 > 0

kai (α, β) = (β, v) = (v, v1) = · · · = (vi, vi+1).Aut h diadikasÐa ìmwc prèpei na termatÐzei met� apì èna peperasmèno pl -

joc bhm�twn kaj¸c h austhr� fjÐnousa akoloujÐa α > β > v > · · · > vi+1 > 0

apoteleÐtai apì fusikoÔc arijmoÔc kai �ra ja up�rqei k�poio n tètoio ¸stevn+1 = 0, opìte (α, β) = (vn, vn+1) = vn. Dhlad o mègistoc koinìc diairè-thc twn α kai β eÐnai to teleutaÐo mh mhdenikì upìloipo pou prokÔptei apì ticdiadoqikèc prohgoÔmenec EukleÐdeiec diairèseic.

Par�deigma. 'Estw α = 356 kai β = 156. Oi EukleÐdeiec diadoqikèc diairè-seic dÐnoun

356 = 156 · 2 + 44

156 = 44 · 3 + 24

44 = 24 · 1 + 20

24 = 20 · 1 + 4

20 = 4 · 5.


Opìte (356, 156) = 4.

Parat rhsh. Qrhsimopoi¸ntac thn prohgoÔmenh diadikasÐa mporoÔme na dì-soume mia �llh apìdeixh tou 1.1.9 wc ex c.

Ston prohgoÔmeno EukleÐdeio Algìrijmo gia touc α kai β, ton fusikì arij-mì n + 1 (gia ton opoÐo jewr same ìti vn+1 = 0) ton onom�zoume m koctou EukleÐdeiou Algìrijmou twn α kai β kai ton sumbolÐzoume me `(α, β), giapar�deigma `(356, 156) = 5.

Efarmìzontac epagwg sto m koc autì, èqoume:An `(α, β) = 1, dhlad an β|α, tìte (α, β) = α ·0+β1. Upojètoume ìti gia

ìlouc touc arijmoÔc α kai β me `(α, β) < n + 1 up�rqoun x, y ∈ Z ètsi ¸ste(α, β) = αx + βy. 'Estw ìti `(α, β) = n + 1, tìte apì thn EukleÐdeia diaÐreshα = βπ + v ja èqoume `(β, v) = n kai sunep¸c up�rqoun x, y ∈ Z ètsi ¸ste(α, β) = (β, v) = βx+ vy. All� v = α−βπ, opìte (α, β) = β(x−πy)+α− y.'Ara up�rqoun x′ = (−y) ∈ Z kai y′ = x − πy ∈ Z tètoioi ¸ste (α, β) =

αx′ + βy′.Gia thn eÔresh enìc x kai enìc y pou ikanopoieÐ thn αx + βy = (α, β) mpo-

roÔme p�li na qrhsimopoi soume ton EukleÐdeio Algìrijmo: Apì tic diadoqikècEukleÐdeiec diairèseic gr�foume ta upìloipa wc

v = α− βπ

v1 = β − vπ1

v2 = v − v1π2

...

vn−2 = vn−4 − vn−3πn−2

vn−1 = vn−3 − vn−2πn−1

vn = vn−2 − vn−1πn.

Opìte èqoume

vn = vn−2 − (vn−3 − vn−2πn−1)πn

= vn−2(1 + πn−1πn) + vn−3(−πn).


Sth sunèqeia antikajistoÔme to vn−2 ap� thn prohgoÔmenh èkfras tou kaipaÐrnoume

vn = (vn−4 − vn−3πn−2)(1 + πn−1πn) + vn−3(−πn)

= vn−3(−πn−2(1 + πn−1πn)− πn) + vn−4(1 + πn−1πn).

SuneqÐzontac, me ton Ðdio trìpo, fj�noume telik� se mia par�stash thc morf cpou jèloume, dhlad (α, β) = vn = αx + βy. Aut h diadikasÐa dÐnei mia nèa(kataskeuastik ) apìdeixh tou 1.1.9.

Gia par�deigma, gia α = 356, β = 156, èqoume

44 = 356− 156 · 224 = 156− 44 · 320 = 44− 24 · 14 = 24− 20 · 1.

Opìte

4 = 24− 20 · 1 = 24− (44− 24 · 1) = 24 · 2− 44 = (156− 44 · 3)2− 44

= 156 · 2 + 44(−7) = 156 + (356− 156 · 2)(−7)

= 356(−7) + 156 · (16).

'Ara oi x = −7 kai y = 16 dÐnoun mia apì tic (�peirec) parast�seic tou (356, 156)

thc morf c αx + βy, x, y ∈ Z.

ShmeÐwsh. Gia thn eÔresh twn x kai y up�rqoun kai �lloi di�foroi trìpoipou elaqistopoioÔn touc qronobìrouc upologismoÔc pou prokÔptoun apì ticdiadoqikèc antikatast�seic twn upoloÐpwn (blèpe Oystein Ore [?] kai S.P. Gla-

sby: Extended Euclid’s algorithm via backward recurrence relations, Math.

Magazine 1999, 72(3), 228–230). EpÐshc axÐzei na anaferjeÐ ed¸ ìti to pl joctwn diairèsewn ston EukleÐdeio algìrijmo gia dÔo jetikoÔc akèraiouc arijmoÔceÐnai mikrìtero pènte forèc apì to pl joc twn dekadik¸n yhfÐwn tou mikrìteroutwn dÔo arijm¸n. Autì eÐnai èna je¸rhma tou Gabriel Lame (1890).


Parat rhsh. 'Olec oi prohgoÔmenec idiìthtec sthn Prìtash 1.1.12 mporoÔnna apodeiqjoÔn efarmìzontac ton algìrijmo tou EukleÐdh. Gia par�deigma,ac apodeÐxoume thn 1.1.12 v), dhlad an (α, β) = 1 kai β|αγ tìte β|γ. Kaj¸c(α, β) = 1 ja èqoume vn = 1. Pollaplasi�zontac ìlec tic EukleÐdeiec diairèseicepÐ γ, upojètontac ìti β|αγ, o β ja diaireÐ ìlouc touc akèraiouc viγ kai �ra kaiton γ = vnγ.

Pr¸toi ArijmoÐ � Jemelei¸dec Je¸rhma thc Arijmhtik c

1.1.13 Orismìc. 'Enac fusikìc arijmìc p > 1 ja kaleÐtai pr¸toc arijmìc apl� pr¸toc an oi mìnoi diairètec tou eÐnai oi ±1 kai ±p. 'Enac fusikìc arijmìcn > 1 pou den eÐnai pr¸toc ja kaleÐtai sÔnjetoc.

Sunep¸c ènac jetikìc akèraioc p > 1 eÐnai pr¸toc an kai mìnon an gia k�jen ∈ Z isqÔei h (p, n) = 1 (p, n) = p, dhlad o p eÐnai sqetik� pr¸toc procton n o p diaireÐ ton n.

Epeid k�je �rtioc arijmìc diaireÐtai me to 2, apì ton orismì prokÔptei ìtiìloi oi pr¸toi ektìc apì ton 2 eÐnai perittoÐ arijmoÐ.

Oi pr¸toi pou eÐnai mikrìteroi tou 100 eÐnai oi ex c 25 arijmoÐ:2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89 kai 97.Shmei¸noume ìti o arijmìc 1 kat� sunj kh den jewreÐtai oÔte pr¸toc oÔte

sÔnjetoc arijmìc. Apodeqìmaste aut th sunj kh diìti diaforetik� den jaisqÔoun basik� apotelèsmata pou ja doÔme amèswc pio k�tw.

1.1.14 L mma. K�je akèraioc > 1 eÐnai ènac pr¸toc arijmìc to ginìmenopr¸twn arijm¸n.

Apìdeixh. 'Estw n ∈ N, n > 1. Upojètoume ìti k�je fusikìc m, 1 < m < n,eÐte eÐnai pr¸toc eÐnai ginìmeno pr¸twn. An o n den eÐnai pr¸toc, dhlad eÐnaisÔnjetoc, tìte èqei ènan diairèth α, 1 < α < n kai n = αβ gia k�poion β ∈ Nme 1 < β < n (giatÐ;). Opìte oi α kai β eÐnai eÐte pr¸toi ginìmeno pr¸twn.'Ara o n an den eÐnai pr¸toc autìc eÐnai ginìmeno pr¸twn. Sunep¸c epagwgik�to zhtoÔmeno isqÔei gia k�je n ∈ N, n > 1.


To 1.1.14 mac lèei ìti oi pr¸toi arijmoÐ eÐnai pollaplasiastik� oi “jemèlioilÐjoi” gia thn kataskeu twn fusik¸n arijm¸n. 'Etsi eÐnai fusikì èna meg�lomèroc thc JewrÐac Arijm¸n na epikentr¸netai sth melèth twn pr¸twn arijm¸n.

Apì ton orismì twn pr¸twn kai apì to 1.1.12 v) prokÔptei �mesa to ex c:

1.1.15 L mma (EukleÐdhc). An α, β ∈ Z kai p eÐnai ènac pr¸toc pou diaireÐto ginìmeno αβ tìte o p diaireÐ ton α o p diaireÐ ton β.

1.1.16 Pìrisma. An p eÐnai pr¸toc kai p|α1α2 · · ·αn, α1, . . . , αn ∈ Z tìtep|αi, gia k�poio i = 1, 2, . . . , n. Eidikìtera, an oi α1, α2, . . . , αn eÐnai pr¸toi tìtep = pi, gia k�poio i.

Apìdeixh. Efarmìzoume epagwg sto n qrhsimopoi¸ntac to prohgoÔmeno l m-ma.

T¸ra ja lème ìti ènac fusikìc arijmìc > 1 analÔetai monadik� se ginìmenopr¸twn paragìntwn an gia dosmènouc pr¸touc p1, p2, . . . , pr kai q1, q2, . . . , qs

tètoiouc ¸sten = p1p2 · · · pr = q1q2 · · · qs

ìsec forèc emfanÐzetai ènac pr¸toc metaxÔ twn p1, p2, . . . , pr tìsec forèc em-fanÐzetai autìc metaxÔ twn q1, q2, . . . , qs (Shmei¸noume ìti ap� autì prokÔpteir = s).

Me aut thn orologÐa apodeiknÔoume t¸ra to Jemelei¸dec Je¸rhma thcArijmhtik c (merikèc forèc autì anafèretai kai wc Je¸rhma Monadik c Para-gontopoÐhshc).

1.1.17 Je¸rhma. K�je fusikìc arijmìc n > 1 analÔetai monadik� se ginì-meno pr¸twn paragìntwn.

1h Apìdeixh. S� aut thn apìdeixh qrhsimopoioÔme to 1.1.16. MporoÔme na upo-jèsoume ìti o n eÐnai sÔnjetoc arijmìc. Upojètoume ìti k�je sÔnjetoc arijmìcmikrìteroc tou n analÔetai monadik� se pr¸touc par�gontec. DeÐqnoume ìtitìte kai o n analÔetai monadik� se pr¸touc par�gontec, opìte to apotèlesmaèpetai apì th majhmatik epagwg sto n.


'Estw loipìn ìtin = p1p2 · · · pr = q1q2 · · · qs

ìpou p1, p2, . . . , pr kai q1, q2, . . . , qs eÐnai pr¸toi kai p1 ≤ p2 ≤ · · · ≤ pr, q1 ≤q2 ≤ · · · ≤ qs. Prèpei na deÐxoume ìti r = s kai pi = qi gia k�je i = 1, 2, . . . , r.

'Estw p o mikrìteroc pr¸toc pou diaireÐ ton n. Tìte lìgw tou 1.1.16,p = pi, gia k�poio i = 1, 2, . . . , r, kai epeid p ≤ p1 ja prèpei p = p1. 'Omoiap = q1, opìte p1 = q1. 'Estw m = n/p. Tìte

m = p2p3 · · · pr = q2 · · · qs.

All� tìte r = s kai pi = qi, i = 2, . . . , r, afoÔ m < n. 'Ara o n analÔetaimonadik� se pr¸touc par�gontec. 'Etsi èqoume deÐxei ìti an ìloi oi sÔnjetoiarijmoÐ m, 1 < m < n analÔontai monadik� se pr¸touc par�gontec, to ÐdioisqÔei kai gia ìlouc touc sÔnjetouc arijmoÔc m, 1 < m < n + 1.

'Ara ìloi oi sÔnjetoi arijmoÐ analÔontai monadik� se pr¸touc par�gontec.

2h Apìdeixh. S� aut thn apìdeixh den ja qrhsimopoi soume to L mma touEukleÐdh. Upojètoume ìti up�rqoun sÔnjetoi arijmoÐ > 1 pou èqoun dÔo dia-foretikèc analÔseic se pr¸touc par�gontec. 'Estw n o mikrìteroc tètoiocarijmìc gia ton opoÐo èqoume

n = p1p2 · · · pr = q1q2 · · · qs

ìpou p1 ≤ p2 ≤ · · · ≤ pr, q1 ≤ q2 ≤ · · · qs.K�je pi eÐnai di�foroc apì k�je qi, diìti diaforetik� an up rqe koinìc pr¸-

toc par�gontac ja paÐrname k�poion n′ < n me thn Ðdia idiìthta pou èqei o n

(pou den mporeÐ na isqÔei lìgw thc upìjeshc ìti o n eÐnai o mikrìteroc m� aut thn idiìthta). 'Estw ìti p1 < q1. JewroÔme to sÔnjeto arijmì

m = p1q2q3 · · · qs.

O fusikìc arijmìc ` = n−m = (q1−p1)q2 · · · qs pou eÐnai mikrìteroc apì ton n

diaireÐtai dia p1, afoÔ p1|n kai p1|m. Opìte o ` analÔetai monadik� se pr¸toucpar�gontec ènac ek twn opoÐwn eÐnai o p1. 'Estw

` = p1t2t3 · · · tk


h an�lush tou ` se pr¸touc par�gontec. An o arijmìc q1− p1 = 1, tìte o ` jaeÐqe dÔo analÔseic se pr¸touc par�gontec, h mia ja perieÐqe ton p1 kai h �llhden ja ton perieÐqe, opìte, epeid ` < n, prèpei q1− p1 6= 1. Sunep¸c mporoÔmena gr�youme ton q1 − p1 wc ginìmeno pr¸twn arijm¸n, èstw

q1 − p1 = h1h2 · · ·ht.

'Etsi èqoume ` = h1h2 · · ·htq2 · · · qs. Aut eÐnai mia an�lush tou ` se pr¸toucpou den perièqei ton p1, afoÔ p1 - q1−p1 kai p1 6= qi, i = 1, 2, . . . , s. 'Opwc ìmwceÐdame o ` èqei kai �llh an�lush se pr¸touc pou perièqei ton p1. Epeid ` < n,autì eÐnai �topo, afoÔ o n eÐnai o mikrìteroc arijmìc me perissìterec thc miacanalÔseic se pr¸touc par�gontec. 'Ara den up�rqei kanènac fusikìc sÔnjetocarijmìc me perissìterec thc miac analÔseic se pr¸touc par�gontec.

ShmeÐwsh. H pr¸th apìdeixh dìjhke apì ton EukleÐdh en¸ h deÔterh apì tonZermelo. Up�rqoun kai �llec apodeÐxeic pou metaxÔ aut¸n xeqwrÐzoun autècpou qrhsimopoioÔn th jewrÐa om�dwn (bl. [?]) thn TopologÐa. (Ja d¸soumeargìtera mia �llh apìdeixh me th qr sh twn akolouji¸n tou Farey).

Parat rhsh. 1. An eÐqame sumperil�bei ton arijmì 1 stouc pr¸touc arij-moÔc, tìte ja èprepe na anadiatup¸noume to prohgoÔmeno je¸rhma epitrèpontacdiaforetikèc paragontopoi seic, gia par�deigma 6 = 2 · 3 = 1 · 2 · 3.

2. Up�rqoun poll� paradeÐgmata diaktulÐwn sthn �lgebra pou den ika-nopoioÔn to je¸rhma thc monadik c paragontopoÐshc. Epeid ed¸ den èqoumeanaferjeÐ se daktulÐouc, ja d¸soume to pio aplì par�deigma. jewroÔme touc�rtiouc akèraiouc 2Z = {. . . ,−4,−2, 0, 2, 4, . . .}. An α, β ∈ 2Z ja lème ìtiβ|α an up�rqei γ ∈ 2Z ètsi ¸ste α = βγ. 'Etsi 2|4 afoÔ 4 = 2 · 2, en¸ to2 den diaireÐ ton eautì tou epeid to 1 /∈ 2Z. 'Ena stoiqeÐo p ∈ 2Z lègetai“pr¸toc” an den up�rqoun α, β ∈ 2Z tètoia ¸ste p = αβ. Gia par�deigma,to 2, to 6, to 14 eÐnai “pr¸toi” sto 2Z. ParathroÔme epÐshc ìti en¸ to 2diaireÐ to 4, to 2 de diaireÐ ènan apì touc par�gontec tou, 4 = 2 · 2. Dhla-d den isqÔei to L mma tou EukleÐdh. ParathroÔme epÐshc ìti den isqÔei toje¸rhma monadik c paragontopoÐhshc, kaj¸c 2 · 18 = 6 · 6 kai oi 2, 6 kai 18


eÐnai “pr¸toi”. Ap� thn �llh meri� k�je mh mhdenikì jetikì stoiqeÐo tou 2ZanalÔetai (paragontopoieÐtai) se “pr¸touc”, kaj¸c to stoiqeÐo 2z analÔetaise pr¸touc akèraiouc: 2z = 2kp1 · · · ps, ìpou p1, . . . , ps eÐnai perittoÐ pr¸toi,opìte èqoume 2z = 2 · · · 2(2p1 · · · ps) ìpou to 2 kai to 2p1 · · · ps eÐnai “pr¸toi”par�gontec tou 2z.

'Ena �llo par�deigma eÐnai to sÔnolo S = {3k + 1/k ∈ N}.An s1, s2 ∈ S, tìte kai s1s2 ∈ S. An α, β ∈ S, tìte lème ìti o β diaireÐ

ton α an α = βγ, gia k�poio γ ∈ S. 'Ena de stoiqeÐo p ∈ S lègetai S-pr¸tocan p > 1 kai gia r > 1, r ∈ S me r|p tìte r = p. Gia par�deigma, oi arijmoÐ4, 7, 10 kai 13 eÐnai S-pr¸toi en¸ o 1 kai o 16 den eÐnai S-pr¸toi. K�je depr¸toc thc morf c 3κ + 1 eÐnai S-pr¸toc ìpwc epÐshc to ginìmeno dÔo pr¸twnthc morf c 3κ + 2 eÐnai S-pr¸toc (afoÔ (3κ + 2)(3κ′ + 2) = 3κ′′ + 1). 'Estw3λ + 1 = p1p2 · · · ps h an�lush se pr¸touc enìc stoiqeÐou tou S. Epeid k�jepr¸toc 6= 3 eÐnai thc morf c 3κ + 1 3κ + 2, k�je pi eÐnai thc morf c 3κ + 1 3κ + 2. Epeid de (3κ + 1)(3κ′ + 2) = 3κ′′ + 2, sthn an�lush tou 3λ + 1 prèpeina up�rqoun �rtio pl joc pr¸twn thc morf c 3κ+2 pou to ginìmenì touc eÐnaiènac S-pr¸toc. 'Ara k�je stoiqeÐo tou S analÔetai se ginìmeno S-pr¸twn.

ParathroÔme ìmwc ìti

100 = 3 · 33 + 1 = 4 · 25 = 10 · 10

ìpou to 4, to 10 kai to 25 eÐnai S-pr¸toi, pou shmaÐnei ìti den èqoume monadik paragontopoÐhsh.

T¸ra k�noume thn paradoq ìti o arijmìc n = 1 eÐnai to “kenì” ginìmenopr¸twn. An de stic analÔseic twn fusik¸n arijm¸n se pr¸touc par�gontecsullèxoume touc Ðsouc pr¸touc se dun�meic pr¸twn tìte to 1.1.17 mporeÐ naxanadiatupwjeÐ wc ex c.

1.1.18 Je¸rhma. K�je fusikìc arijmìc n > 0 gr�fetai monadik� sth morf

n = pα11 pα2

2 · · · pακκ

ìpou p1, p2, . . . pκ eÐnai pr¸toi me p1 < p2 < · · · < pκ kai αi ≥ 0, i = 1, 2, . . . , κ.


Merikèc forèc mac boleÔei na gr�foume thn an�lush tou n se ginìmenopr¸twn kai wc

n =∏

p∈P

pαp

ìpou P eÐnai to sÔnolo ìlwn twn pr¸twn, k�je αp ≥ 0 kai mìno èna peperasmènopl joc ekjet¸n αp eÐnai 6= 0.

Oi ekjètec αp pou emfanÐzontai sthn prohgoÔmenh paragontopoÐhsh tou n

sun jwc sumbolÐzontai me vp(n) kai qarakthrÐzontai ap� thn ex c idiìthta

αp = vp(n) ⇐⇒ pαp |n kai pαp+1 - n.

Gia par�deigma, èqoume 600 = 24 ·3−5, opìte v2(600) = 4, v3(600) = v5(600) =

1 kai vp(600) = 0 gia k�je pr¸to p 6= 2, 3, 5.EÐnai fanerì ìti k�je n ∈ N orÐzetai monadik� apì touc ekjètec vp(n).H apeikìnish vp : N→ N èqei tic ex c aplèc idiìthtec.

1.1.19 Prìtash. Gia k�je m,n ∈ N, m,n ≥ 1, isqÔoun ta ex c:

i) vp(n) = 0, gia k�je pr¸to p an kai mìnon an n = 1

ii) vp(mn) = vp(m) + vp(n)

iii) m|n an kai mìnon an vp(m) ≤ vp(n), gia k�je pr¸to p. Sunep¸c h isìthtaisqÔei an kai mìnon an m = n.

iv) δ = (m, n) an kai mìnon an vp(δ) = min{vp(n), vp(m)}, gia k�je pr¸to p.

v) ε = (m,n) an kai mìnon an vp(ε) = max{vp(n), vp(m)} gia k�je pr¸to p.

Apìdeixh. ApodeiknÔoume thn iv), oi upìloipec af nontai wc ask seic. 'Estwδp = min{vp(m), vp(n)}. Gia k�je p pr¸to, èqoume pδp |m kai pδp |n. 'Ara

pδp |(m,n). Opìte∏

pδp |(m,n). An tan 1 < α =(m, n)∏

pδp, tìte α|(m,n) kai

an p eÐnai ènac pr¸toc diairèthc tou α tìte p|m kai p|n. Autì shmaÐnei ìtipδp+1|(m,n). All� tìte pvp(n)+1|n kai pvp(m)+1|m pou eÐnai adÔnaton. 'Araα = 1.


Shmei¸noume ìti h idiìthta 1.1.19 ii) eÐnai h Ðdia m� aut pou isqÔei stouclog�rijmouc.

Parat rhsh. H paragontopoÐhsh twn jetik¸n akeraÐwn se pr¸touc mporeÐna epektajeÐ kai stouc arnhtikoÔc akeraÐouc jètontac ±1 emprìc apì to ginì-meno. EpÐshc mporeÐ na epektajeÐ kai stouc mh mhdenikoÔc rhtoÔc, epitrèpontacoi ekjètec na eÐnai arnhtikoÐ. 'Etsi mporoÔme na epekteÐnoume thn apeikìnish vp

sto Q−{0} jètontac vp(−n) = vp(n) kai vp(n/m) = vp(n)−vp(m) gian

m6= 0,

(ed¸ prèpei na deiqjeÐ ìti h apeikìnish eÐnai kal� orismènh anex�rthta apì thnpar�stash tou kl�smatoc

n

m).

ParadeÐgmata.

1. Qrhsimopoi¸ntac to Pìrisma 1.1.11 eÐqame deÐxei ìti o arijmìc√

2 eÐnai�rrhtoc. To apotèlesma autì mporeÐ t¸ra na deiqjeÐ qrhsimopoi¸ntac tojemelei¸dec je¸rhma thc arijmhtik c: 'Estw

√2 =

α

β, ìpou (α, β) = 1,

α, β ∈ N. Tìte 2β2 = α2. An α = pα11 pα2

2 · · · pαss kai β = qβ1

1 qβ22 · · · qβt

t ,tìte o α2 kai o β2 analÔontai se �rtio pl joc pr¸twn paragìntwn en¸ o2β2 se perittì. 'Ara den mporeÐ na isqÔei 2β2 = α2. Me ton Ðdio isqurismìapodeiknÔetai ìti gia k�je pr¸to p kai �rtio n o arijmìc n

√p eÐnai �rrhtoc.

All� epÐshc kai autì perilamb�netai wc eidik perÐptwsh tou ex c genikoÔapotelèsmatoc: An o α ∈ N, den eÐnai h n-iost dÔnamh enìc fusikoÔ arij-moÔ, gia k�poio n ∈ N, tìte o n

√α eÐnai �rrhtoc. Pr�gmati upojètontac

ìti eÐnai rhtìc èstw n√

α =m

n, (m,n) = 1, tìte α =

mκ

nκ αnκ = mκ.

Opìte efarmìzontac thn apeikìnish vp èqoume κvp(m) = vp(α) + κvp(n).Sunep¸c vp(n) ≤ vp(m). 'Ara, lìgw thc 1.1.19 iii), prèpei n|m pou eÐnai�topo.

2. 'Estw α, β ∈ N me αβ = γ2, gia k�poio γ ∈ N. Tìte up�rqoun γ1, γ2 ∈ Nètsi ¸ste

α

(α, β)= γ2

1 kaiβ

(α, β)= γ2

2 .

Pr�gmati, èstwα

(α, β)= pα1

1 · · · pass kai

β

(α, β)= qβ1

1 · · · qβκκ oi analÔseic


se pr¸touc. Epeid oα

(α, β)eÐnai sqetik� pr¸toc proc ton

β

(α, β)ta

sÔnola {p1, . . . , ps} kai {q1, . . . , qκ} eÐnai xèna metaxÔ touc. 'Eqoume depα11 · · · pαs

s qβ11 · · · qβκ

κ =( γ

(α, β)

)2, opìte ta αi kai βi prèpei na eÐnai �rtioi

arijmoÐ kai sunep¸c oiα

(α, β)kai

β

(α, β)eÐnai tetr�gwna akeraÐwn.

3. To ginìmeno tri¸n diadoqik¸n fusik¸n arijm¸n den mporeÐ na eÐnai miadÔnamh enìc fusikoÔ arijmoÔ. Pr�gmati, èstw ìti o akèraioc (n−1)n(n+

1) = (n2 − 1)n eÐnai h m-ost dÔnamh enìc akeraÐou. All� to n2 − 1

kai to n eÐnai sqetik� pr¸toi arijmoÐ. Apì to jemelei¸dec je¸rhma thcArijmhtik c ja prèpei tìte to n2− 1 kai to n2 na eÐnai m-iostèc dun�meicakeraÐwn. (giatÐ;). Epeid ìmwc to n2 − 1 kai to n2 eÐnai diadoqikoÐ autìden mporeÐ na isqÔei (giatÐ;).

4. An α, β ∈ N tètoioi ¸ste α|β2, β2|α3, α3|β4, β4|α5, . . . . Tìte α = β.'Estw ìti

α = pα11 · · · pαs

s kai β = qβ11 · · · qβt

t

oi analÔseic twn α kai β se pr¸touc. Mac dÐnetai ìti gia n = 1, 2, . . .

isqÔei

kaip(2n−1)α1

1 · · · p(2n−1)αss

∣∣q2nβ11 · · · q2nβt

t

q2nβ11 · · · q2nβt

t

∣∣p(2n+1)α1

1 · · · p(2n+1)αss .

'Ara to pi|β, gia i = 1, . . . , s kai qj |α gia j = 1, . . . , t. Opìte s = t kai memia kat�llhlh arÐjmhsh twn deikt¸n èqoume pi = qi, i = 1, . . . , s.

'Ara αi ≤ 2n

2n− 1βi kai βi ≤ 2n + 1

2nαi. Sunep¸c αi − βi ≤ βi

2n− 1kai

βi − αi ≤ αi

2nkaj¸c n →∞ prokÔptei ìti αi = βi.

To Pl joc twn Pr¸twn

EÐnai fusikì na anarwthjoÔme pìsoi pr¸toi arijmoÐ up�rqoun. Thn ap�nthshs� autì to er¸thma èdwse o EukleÐdhc prin perÐpou 2300 qrìnia (Prìtash 20,9o biblÐo, StoiqeÐa tou EukleÐdh).


1.1.20 Je¸rhma (EukleÐdhc). To pl joc twn pr¸twn arijm¸n eÐnai �peiro.

Apìdeixh (EukleÐdhc). Upojètoume ìti autì to pl joc den eÐnai �peiro kaièstw ìti ìloi oi pr¸toi eÐnai oi 2 = p1, p2, . . . , pn. JewroÔme ton arijmìN = p1p2 · · · pn + 1 ≥ 3. Epeid N > pj , gia 1 ≤ j ≤ n, o N den eÐnaipr¸toc kai kaj¸c N > 1, lìgw tou 1.1.17, o N diaireÐtai di� enìc pr¸tou, èstwton pκ. All� tìte o pκ ja prèpei na diaireÐ kai to 1 = N − (N − 1) pou eÐnai�topo.

Parathr seic.

1. H prohgoÔmenh apìdeixh paramènei h Ðdia an antÐ tou N jewr soume tonarijmì p1p2 · · · pn − 1.

2. H apìdeixh tou 1.1.20 eÐnai mia eidik perÐptwsh thc ex c apìdeixhc pouofeÐletai ston T.J. Stieljes. 'Estw A to ginìmeno opoiond pote r apìtouc n pr¸touc pi, 1 ≤ r ≤ n, pou jewr same sthn apìdeixh kai B =

p1p2 · · · pn/A. Tìte o A + B den diaireÐtai apì kanènan pi, i = 1, . . . , n.Epeid A + B > 1, o A + B prèpei na èqei ènan pr¸to diairèth di�forotwn pi, 1 ≤ i ≤ n.

3. An pn eÐnai o n-iostìc pr¸toc, tìte ìpwc sthn apìdeixh tou EukleÐdh,gia k�poio m > n, o m-iostìc pr¸toc pm ja diaireÐ ton 2 · 3 · · · pn + 1

kai eÐnai pm ≥ pn+1. Sunep¸c m� autì ton trìpo, apì ènan pr¸to p

kajorÐzetai toul�qiston ènac �lloc pr¸toc megalÔteroc apì ton p poueÐnai diairèthc tou 2 · 3 · · · p + 1. Gia par�deigma, o 7, o 31 kai o 211

kajorÐzontai wc diairètec antÐstoiqa tou 2 ·3+1 = 7, tou 2 ·3 ·5+1 = 31

kai tou 2 · 3 · 5 · 7 + 1 = 211. En¸ o 59 kai o 509 kajorÐzontai wcdiairètec tou 2 · 3 · 5 · 7 · 11 · 13 · 17 + 1 = 59 · 509. Paramènei anap�nthtoto er¸thma an up�rqoun �peiroi to pl joc pr¸toi p gia touc opoÐouc oiarijmoÐ 2 ·3 · · · p+1 eÐnai pr¸toi an up�rqoun �peiroi to pl joc sÔnjetoiarijmoÐ aut c thc morf c. Sthn ptuqiak tou ergasÐa o Alan Borning (J.

Mathematics of Computations 1972) k�nontac upologismoÔc br ke ìti


metaxÔ ìlwn twn pr¸twn p ≤ 307 mìno gia touc p = 2, 3, 5, 7, 11 kai 31

oi arijmoÐ 2 · 3 · · · p + 1 eÐnai pr¸toi, en¸ gia p = 3, 5, 11, 13, 41 kai 89 oiarijmoÐ 2 · 3 · · · p− 1 eÐnai pr¸toi.

4. 'Opwc eÐdame, an o pn eÐnai o n-iostìc pr¸toc tìte, gia k�poio m > n, o

pm diaireÐ tonn∏

i=1pi + 1, opìte

pn+1 ≤ pm ≤n∏

i=1

pi + 1 ≤ pnn + 1.

Ap� aut th sqèsh kai th majhmatik epagwg prokÔptei ìti gia k�jen ≥ 1 isqÔei pn ≤ 22n−1 me pn < 22n−1 , gia n > 1. Sunep¸c up�rqountoul�qiston n pr¸toi pou eÐnai mikrìteroi tou 22n−1 . Pr�gmati, èqoumep1 ≤ 2 kai upojètoume ìti p2 ≤ 22, p3 ≤ 24, . . . , pn ≤ 22n−1 . Tìtepn+1 ≤ p1 · · · pn+1 ≤ 21+2+···+2n−1

+1 = 22n−1+1 < 22n−1+22n−1 = 22n .Ap� autì sumperaÐnoume ìti gia n ≥ 1 up�rqoun toul�qiston n+1 pr¸toipou eÐnai mikrìteroi tou 22n .

5. Jètoume n1 = 2 kai epagwgik� orÐzoume touc arijmoÔc nκ wc nκ+1 =

n2κ−nκ +1. Opìte nκ+1 = n1 · · ·nκ +1. An m eÐnai ènac koinìc diairèthc

tou ns kai n` (èstw s < `), tìte o m diaireÐ ton n1 · · ·ns · · ·n`−1 kai tonn` = n1 · · ·n`−1 + 1, sunep¸c m = 1. 'Ara an� dÔo oi nκ eÐnai sqetik�pr¸toi metaxÔ touc. Epeid oi arijmoÐ nκ (pou eÐnai �peiroi se pl joc)an� dÔo eÐnai sqetik� pr¸toi metaxÔ touc kai o kajènac èqei ènan pr¸todiairèth, up�rqoun �peiroi se pl joc pr¸toi. Aut h apìdeixh eÐnai miadiaforopoÐhsh tou sulogismoÔ thc apìdeixhc tou EukleÐdh all� ìmwcupodeiknÔei ìti s� autì to sulogismì ekeÐno pou paÐzei shmantikì rìloeÐnai ìti oi pi eÐnai pr¸toi an� dÔo kai ìqi tìso ìti autoÐ eÐnai pr¸toiarijmoÐ.

S� aut thn idèa sthrÐzetai kai h epìmenh apìdeixh tou 1.1.20 pou ofeÐletaiston G. Polya. JewroÔme tou arijmoÔc

Fn = 22n+ 1, n ∈ N


kai apodeiknÔoume ìti an� dÔo autoÐ eÐnai sqetik� pr¸toi. Pr¸ta deÐqnoumeìti isqÔei h sqèsh

Fn = F0F1 · · ·Fn−1 + 2.

Pr�gmati gia n = 1 profan¸c isqÔei. Epagwgik�, upojètontac ìti Fn =

F0F1 · · ·Fn−1 + 2, èqoume

F0F1 · · ·Fn = (Fn − 2)Fn = (22n − 1)(22n+ 1) = 22n+1 − 1 = Fn+1 − 2,

ìpwc apaiteÐtai.

T¸ra an δ eÐnai ènac diairèthc twn Fk kai F` (èstw k < `), tìte o δ eÐnaidiairèthc kai tou F` − F0F1 · · ·Fk · · ·F`−1 = 2. 'Ara o δ eÐnai o 1 o 2.All� epeid oi arijmoÐ tou Fermat eÐnai perittoÐ, prèpei δ = 1. Sunep¸c,gia k�je n ∈ N, up�rqoun toul�qiston n+1 diakekrimènoi pr¸toi arijmoÐ,afoÔ k�je Fi, i ≤ n, èqei èna pr¸to diairèth (o opoÐoc den diaireÐ kanènan�llo Fj , j 6= i, j ≤ n). Epeid up�rqoun �peiroi arijmoÐ tou Fermat,up�rqoun �peiroi pr¸toi arijmoÐ. (Mia pio sÔntomh apìdeixh eÐnai h ex c.Upojètoume ìti m > n, kai èstw m = n + κ, κ > 0. Jètoume x = 22n ,opìte

Fn+κ − 2Fn

=22n2κ − 122n + 1

=x2κ − 1x + 1

= x2κ−1 − x2κ−2 + · · · − 1 ∈ Z.

Autì shmaÐnei ìti Fn|Fn+κ − 2. Opìte an δ|Fn kai δ|Fn+κ ja prèpei δ|2.All� oi Fn kai Fn+κ eÐnai perittoÐ, �ra δ = 1.)

AutoÐ oi arijmoÐ onom�zontai arijmoÐ tou Fermat, kai fèroun aut thnonomasÐa epeid o Fermat se mia epistol tou proc ton Pascal kai proc�llouc ègrafe ìti autoÐ oi arijmoÐ eÐnai pr¸toi. O Fermat diatÔpwse aut thn eikasÐa apì to gegonìc ìti oi pr¸toi 5 tètoioi arijmoÐ eÐnai pr¸toi:F0 = 3, F1 = 5, F2 = 17, F3 = 257, F4 = 65537. O epìmenoc arijmìctou Fermat F5 = 4294967297 eÐnai arket� meg�loc kai thn epoq ekeÐnh tan dÔskolo na paragontopoihjeÐ se pr¸touc. 100 qrìnia met� o Euler

to 1739 apèdeixe ìti k�je diairèthc enìc arijmoÔ tou Fermat prèpei naeÐnai thc morf c 2n+1k + 1. Sunep¸c, gia n = 5 ènac pr¸toc par�gontac


tou F5 prèpei na eÐnai thc morf c 64k + 1. 'Etsi eÔkola brèjhke ìti o641 = 26 · 10 + 1 diaireÐ ton F5 kai �ra h eikasÐa tou Fermat den isqÔei.To 1877 o J. Pepin apèdeixe ìti o Fn eÐnai pr¸toc an kai mìnon an autìcdiaireÐ ton 3

Fn−12 + 1. EpÐshc o Fn, n ≥ 1, eÐnai pr¸toc an kai mìnon an o

monadikìc trìpoc pou mporeÐ na grafteÐ o Fn wc �jroisma dÔo tetrag¸nwneÐnai o profan c:

Fn = (22n−1)2 + 22.

Paramènei anap�nthto to er¸thma an k�je arijmìc tou Fermat den diai-reÐtai apì to tetr�gwno enìc arijmoÔ.

Up�rqoun kai �llec anagkaÐec kai ikanèc sunj kec gia na elègqoume an oFn eÐnai pr¸toc, ìmwc den up�rqei èwc s mera ènac genikìc kanìnac pou jaodhgoÔse se mia kajoristik ap�nthsh sto anap�nthto er¸thma: EÐnai oF4 o megalÔteroc pr¸toc arijmìc tou Fermat; dhlad gia n > 4, o Fn eÐnaisÔnjetoc; an autì den isqÔei, up�rqoun �peiroi to pl joc pr¸toi arijmoÐtou Fermat; up�rqoun �peiroi to pl joc sÔnjetoi arijmoÐ tou Fermat;

H ap�nthsh s� aut� ta erwt mata eÐnai shmantik giatÐ, ìpwc apèdeixeto 1801 o Gauss sto gnwstì Disquisitiones Arithmeticae, up�rqei miaaxishmeÐwth sqèsh metaxÔ twn EukleÐdeiwn kataskeu¸n (dhlad , me thqr sh kanìna kai diab th) twn kanonik¸n polug¸nwn kai twn arijm¸ntou Fermat. Sugkekrimèna, o Gauss èdeixe ìti an to pl joc twn pleur¸nenìc kanonikoÔ polug¸nou eÐnai thc morf c 2κFm1 · · ·Fmr , ìpou κ ≥ 0,r ≥ 0 kai Fmi eÐnai diakekrimènoi pr¸toi arijmoÐ tou Fermat, tìte autì topolÔgwno mporeÐ na kataskeuasjeÐ me kanìna kai diab th. To antÐstrofoautoÔ tou jewr matoc apedeÐqjei to 1837 apì ton P.L. Wantzel, an kai oGauss sto Disquisions Arithmeticae isqurÐzetai ìti isqÔei (qwrÐc ìmwc nato apodeiknÔei). Oi ArqaÐoi 'Ellhnec gn¸rizan thn EukleÐdeia kataskeu mìno gia to isìpleuro trÐgwno kai to kanonikì pent�gwno �ra kai giak�je kanonikì polÔgwno pou èqei n pleurèc, ìpou n = 2κ, 2κ · 3, 2κ ·5 kai 2κ · 15.∗ 'Etsi met� apì 2000 qrìnia, me to je¸rhma tou Gauss,

∗K�je gwnÐa mporeÐ na diqotomhjeÐ. Sunep¸c an èna n-gwno eÐnai kataskeu�simo tìte


apant jhke pl rwc èna apì ta shmantikìtera erwt mata twn ArqaÐwn:poia kanonik� polÔgwna eÐnai EukleÐdeia kataskeu�sima. SÔmfwna me toje¸rhma tou Gauss kai èqontac upìyin ìti oi mìnoi gnwstoÐ èwc s merapr¸toi arijmoÐ tou Fermat eÐnai oi 3, 5, 17, 257 kai 65537, to pl joc twngnwst¸n tètoiwn polug¸nwn pou èqoun perittì pl joc pleur¸n (ed¸perilamb�noume kai autì pou èqei mia pleur�) isoÔtai me to pl joc twndiairet¸n tou arijmoÔ 1 · 3 · 5 · 17 · 257 · 65537 kai autì to pl joc eÐnai 32.(Shmei¸noume ìti 232 − 1 = 1 · 3 · 5 · 17 · 257 · 65537).

EpÐshc apì to je¸rhma tou Gauss prokÔptei ìti genik� den eÐnai dunat h triqotìmhsh miac gwnÐac, afoÔ diaforetik�, gia par�deigma, triqoto-m¸ntac tic gwnÐec (60o) enìc isìpleurou trig¸nou ja tan EukleÐdeiakataskeu�simo kai to kanonikì polÔgwno 9(= 21 + 1)2 pleur¸n.

Lègetai, ìti apì to je¸rhma autì o Gauss apof�sise na afier¸sei th zw tou sth melèth twn majhmatik¸n. 'Htan de polÔ uper fanoc gi� autì toapotèlesma kai z thse na topojethjeÐ èna 17-gwno ston t�fo tou. HepijumÐa tou aut den ekplhr¸jhke sto Gottingen ìpou et�fh. Up�r-qei ìmwc sto mnhmeÐo tou pou èqei anegerjeÐ sth gennèthr� tou pìlh toBrunswick.

H idèa sthn apìdeixh tou Jewr matoc tou EukleÐdh mporeÐ epÐshc na efarmo-sjeÐ se orismènec peript¸seic gia na deÐqnoume ìti to pl joc miac sugkekrimènhcmorf c pr¸twn arijm¸n eÐnai �peiro. Gia par�deigma, ènac perittìc pr¸toc arij-mìc eÐnai thc morf c 4k +1 thc morf c 4k +3. Gia to pl joc twn pr¸twn thc

kai to kanonikì 2κn-gwno eÐnai kataskeu�simo. EpÐshc an dÔo kanonik� polÔgwna me n1 kain2 pleurèc, ìpou (n1, n2) = 1, eÐnai kataskeu�sima tìte kai to kanonikì polÔgwno me n1n2

pleurèc eÐnai kataskeu�simo. Pr�gmati, up�rqoun akèraioi x1, x2 tètoioi ¸ste

n1x1 − n2x2 = 1

360

n1n2= x1

360

n2− x2

360

n1.

Opìte h gwnÐa 360

n1n2eÐnai kataskeu�simh. P.q. to kanonikì 15-gwno eÐnai kataskeu�simo

afoÔ to isìpleuro trÐgwno kai to pent�gwno eÐnai kataskeu�sima.


morf c 4k+3 h mèjodo tou EukleÐdh mporeÐ na efarmosjeÐ wc ex c. Upojètoumeìti up�rqoun mìno peperasmènou pl jouc tètoioi pr¸toi p1 = 3, p2 = 7, . . . , pn

kai jewroÔme ton arijmì N = 4p1 · · · pn+3. Autìc o arijmìc den mporeÐ na eÐnaipr¸toc (afoÔ N 6= pj kai ap� thn upìjesh oi pj exantloÔn ìlouc touc pr¸toucthc morf c 4k + 3). 'Ara autìc prèpei na èqei pr¸touc diairètec. An ìloi autoÐoi diairètec èqoun th morf 4k+1 tìte kai o N prèpei na eÐnai aut c thc morf ckaj¸c (4k1 + 1)(4k2 + 1) = 4(4k1k2 + k1 + k2) + 1. All� o N den èqei aut th morf kai �ra toul�qiston ènac apì touc pr¸touc diairètec tou, èstw o p,prèpei na eÐnai thc morf c 4n + 3. Epeid p|N , o p eÐnai di�foroc twn pj , diìtidiaforetik� o p ja èprepe na eÐnai 1. 'Ara up�rqoun �peiroi pr¸toi thc morf c4k + 3.

Kaj¸c gnwrÐzoume ìti to pl joc ìlwn twn pr¸twn eÐnai �peiro kai ìti toÐdio isqÔei kai gia ekeÐnouc thc morf c 4k+3, to pl joc twn pr¸twn thc morf c4k+1 mporeÐ na eÐnai �peiro peperasmèno. 'Omwc den mporoÔme na apofanjoÔmepoio eÐnai autì to pl joc qrhsimopoi¸ntac thn Ðdia idèa me aut pou efarmìsamegia touc pr¸touc thc morf c 4k + 3, kaj¸c tètoioi arijmoÐ mporoÔn na eÐnaiginìmeno pr¸twn thc morf c 4k +3, p.q. 21 = 3 · 7. Argìtera ja deÐxoume, statetragwnik� upìloipa, ìti kai autì to pl joc twn pr¸twn eÐnai �peiro. EpÐshcshmei¸noume ìti h idèa tou EukleÐdh mporeÐ na efarmosjeÐ gia touc pr¸touc thcmorf c 6k+5 ìpwc kai gia touc pr¸touc thc morf c 3k+2. Den mporoÔme ìmwcna mimhjoÔme aut thn idèa gia touc pr¸touc thc morf c 8k + 7. ParathroÔmet¸ra ìti ìloi autoÐ oi pr¸toi eÐnai ìroi arijmhtik¸n proìdwn αn + β, n ∈ N.An o m.k.d(α, β) eÐnai 6= 1 tìte ìloi oi ìroi miac tètoiac akoloujÐac diairoÔntaidia tou (α, β) kai �ra kanènac ap� autoÔc den eÐnai pr¸toc. An (α, β) = 1,tìte up�rqoun �peiroi to pl joc sÔnjetoi arijmoÐ thc morf c αn + β, afoÔup�rqoun �peiroi n me (n, β) 6= 1. All� s� aut thn perÐptwsh de mporoÔme naapofanjoÔme, mimoÔmenoi th mèjodo tou EukleÐdh (ìpwc eÐdame p.q. gia α = 8,β = 7), ìti up�rqoun �peiroi to pl joc ìroi miac tètoiac akoloujÐac pou eÐnaipr¸toi arijmoÐ. 'Omwc o Lejeune-Dirichlet (1805�1859) to 1837, sthrizìmenocse proqwrhmènec mejìdouc thc majhmatik c an�lushc, apèdeixe ìti pr�gmati


isqÔei:

1.1.21 Je¸rhma (Dirichlet). K�je arijmhtik prìodoc αn+β me (α, β) = 1,perilamb�nei �peiro pl joc ìrwn pou eÐnai pr¸toi arijmoÐ.

'Eqontac upìyin autì to apotèlesma, dhmiourgeÐtai eÔloga to er¸thma: EÐnaito je¸rhma tou Dirichlet mia eidik perÐptwsh enìc eurÔterou fainomènou; dh-lad parousi�zetai to Ðdio fainìmeno se mia genikìterh oikogèneia ekfr�sewn;Diaisjhtik� anamenìtan ìti autì èprepe na sumbaÐnei, dhlad episteÔeto ìti,ìpwc gia tic arijmhtikèc proìdouc, ja tan dunatìn na mporoÔsame na apodeÐ-xoume ìti mia sun�rthsh pou orÐzetai me aplì trìpo dÐnei �peiro pl joc pr¸twn.Dustuq¸c ìmwc mia tètoia diaÐsjhsh tan esfalmènh. Gia par�deigma, gia p�-ra poll� qrìnia paramènei anoiktì prìblhma an oi arijmoÐ thc morf c n2 + 1,n ∈ N, perilamb�noun �peiro pl joc pr¸twn, (ìpwc èqoun eik�sei oi G. Hardy

kai J. Littlewood). To 1978 o H. Iwaniec apèdeixe ìti up�rqoun �peiroi topl joc arijmoÐ thc morf c n2 + 1 pou eÐnai eÐte pr¸toi ginìmeno dÔo pr¸-twn. Mia �llh oikogèneia pr¸twn eÐnai autoÐ thc morf c 2p−1, ìpou p pr¸toc.(Shmei¸noume ìti apì to Par�deigma 4 met� thn Prìtash 1.1.12, prokÔptei ìtian o 2α − 1 eÐnai pr¸toc tìte anagkastik� o α eÐnai pr¸toc. To antÐstrofoden isqÔei, p.q. 211 − 1 = 2047 = 23 · 89). AutoÐ onom�zontai pr¸toi arijmoÐtou Mersenne. EÐnai �gnwsto mèqri s mera an up�rqoun �peiroi tètoioi pr¸toi.Shmei¸noume ìti o megalÔteroc pr¸toc arijmìc pou eÐnai gnwstìc mèqri s meraeÐnai o 44os pr¸toc arijmìc tou Mersenne. Brèjhke to Septèmbrio tou 2006eÐnai o 232582657 − 1. Autìc èqei 9808358 dekadik� yhfÐa. O 43os pr¸toc touMersenne brèjhke to Dekèmbrio tou 2005 kai eÐnai o 230402457 − 1. Autìc èqei9152052 dekadik� yhfÐa, en¸ o prohgoÔmenoc pr¸toc èqei 7816230 yhfÐa kaieÐnai o 225964951 − 1.

Thc Ðdiac fÔshc er¸thma pou paramènei ìmwc anap�nthto afor� touc dÐdu-mouc pr¸touc. 'Enac pr¸toc arijmìc onom�zetai dÐdumoc pr¸toc an h diafor�tou epìmenou tou prohgoÔmenou ap� autìn pr¸tou arijmoÔ eÐnai 2, dhlad o p

eÐnai dÐdumoc pr¸toc an o p + 2 o p− 2 eÐnai pr¸toc. Sunep¸c autoÐ mporoÔnna grafoÔn se zeÔgh, gia par�deigma, (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), . . . .


'Oloi oi dÐdumoi ektìc apì touc 3 kai 5 eÐnai thc morf c 6k±1, o de megalÔterocap� autoÔc pou eÐnai gnwstìc mèqri s mera (upologÐsjhke ton AÔgousto tou2005) eÐnai o

100314512544015 · 217196 − 1

kai èqei 51779 yhfÐa.Sqetikì me to prìblhma thc Ôparxhc �peirou pl jouc didÔmwn pr¸twn eÐnai

to prìblhma thc Ôparxhc �peirou pl jouc pr¸twn thc morf c 2p + 1, ìpou p

pr¸toc. Sqetikèc anaforèc gia ta erwt mata aut�, all� kai gia poll� �llatou Ðdiou tÔpou, up�rqoun se di�fora biblÐa thc JewrÐac Arijm¸n all� kai sto“Internet”.

AxÐzei na shmei¸soume ed¸ ìti arketoÐ majhmatikoÐ èqoun prospaj sei nabroun ènan aplì tÔpo pou na dÐnei wc timèc tou ìlouc touc pr¸touc ( akìmhperissìtero mìno touc pr¸touc). 'Ena apì ta ikanopoihtik� apotelèsmata sthnkateÔjunsh aut eÐnai tou W.H. Mills. Autìc to 1947 apèdeixe ìti up�rqei ènac�rrhtoc pragmatikìc arijmìc a > 1 tètoioc ¸ste o n-iostìc pr¸toc pn isoÔtaime [a3n

], ìpou me [x] sumbolÐzoume to megalÔtero akèraio pou eÐnai mikrìterocapì to x. Mèqri s mera den èqei kajorisjeÐ ènac sugkekrimènoc tètoioc arijmìcα. EpÐshc to 1970 apì to axioshmeÐwto apotèlesma tou Yuri Matijasevich pouèluse to 10o prìblhma tou Hilbert (sumplhr¸nontac tic ergasÐec twn Martin

Davis, Hilary Putman kai thc Julia Robinson) proèkuye ìti up�rqoun polu¸-numa me akèraiouc suntelestèc twn opoÐwn ìlec oi jetikèc timèc pou paÐrnounstouc fusikoÔc arijmoÔc eÐnai akrib¸c oi pr¸toi arijmoÐ. 'Ena tètoio polu¸nu-mo kajorÐsjhke to 1976 apì touc J. Jones, D. Sato, H. Wada kai D. Wiens.Autì to polu¸numo eÐnai 25oυ bajmoÔ kai 26 metablht¸n.

EÐnai eukairÐa ed¸ na apodeÐxoume to ex c je¸rhma.Je¸rhma (Goldbach). Den up�rqei èna mh-stajerì polu¸numo f(x) me akè-raiouc suntelestèc pou ìlec oi timèc tou stouc fusikoÔc arijmoÔc na eÐnai pr¸toiarijmoÐ.

Apìdeixh. 'Estw ìti to je¸rhma den isqÔei kai

f(x) = αrxr + · · ·+ α1x + α0, αi ∈ Z


eÐnai èna polu¸numo bajmoÔ ≥ 1 pou oi timèc touc stouc fusikoÔc eÐnai pr¸toiarijmoÐ.

'Estw ènac pr¸toc arijmìc kai gia k�poio n ∈ N, f(n) = p. Tìte gia k�jek ∈ Z, oi akèraioi arijmoÐ f(n + kp) diairoÔntai dia p, afoÔ èqoume

f(n + kp) =αr(n + kp)r + · · ·+ α1(n + kp) + α0 = αr

r∑

i=0

(r

i

)ni(kp)r−i

+ αr−1

r−1∑

i=0

(r − 1

i

)ni(kp)r−1−i + · · ·+ α1(n + kp) + α0

=αrnr + · · ·+ α1n + α0 + αr

r−1∑

i=0

(r

i

)ni(kp)r−i + · · ·+ α1kp

=f(n) + pg(k) = p(1 + g(k)),

ìpou g(k) eÐnai èna polu¸numo tou k me akèraiouc suntelestèc. Epeid p|f(n+

kp), apì thn upìjesh ja prèpei f(n+ kp) = p gia ìla ta k. All� tìte autì jas maine ìti to polu¸numo F (k) = f(n + kp) − p bajmoÔ r èqei �peiro pl jocriz¸n. Autì ìmwc de mporeÐ na isqÔei kaj¸c apì to Jemelei¸dec Je¸rhmathc 'Algebrac èna polu¸numo bajmoÔ r de mporeÐ na èqei perissìterec apì r

rÐzec.

Anafèroume epÐshc ìti eÐnai akìma anap�nthto to er¸thma an up�rqei ènapolu¸numo deutèrou bajmoÔ miac metablht c pou na paÐrnei stouc akèraiouc�peiro pl joc tim¸n oi opoÐec na eÐnai pr¸toi arijmoÐ. Autì den eÐnai gnwstìan isqÔei oÔte gia mia eidik perÐptwsh tètoiou poluwnÔmou.

Grammikèc Diofantikèc Exis¸seic

H lèxh “diofantikèc” proèrqetai apì to ìnoma tou 'Ellhna Diìfantou thc Ale-x�ndreiac o opoÐoc èzhse ton trÐto ai¸na m.Q. kai tan o pr¸toc pou melèthseme susthmatikì trìpo tic akèraiec lÔseic exis¸sewn. To diaswjèn èrgo touDiìfantou “Ta Arijmhtik�” (prìkeitai gia ta èxi biblÐa apì ìlo to èrgo tou)dhmosieÔjhkan se biblÐo apì ton Bachet de Mezitiac to 1621 pou melèthse o


Pierre de Fermat (1608�1665). To biblÐo autì eÐnai s mera gnwstì kaj¸c staperij¸ria twn selÐdwn enìc antitÔpou tou o Fermat eÐqe diatup¸sei pollèc apìtic idèec tou. Autèc oi idèec apotèlesan thn arq thc diamìrfwshc thc JewrÐacArijm¸n wc enìc xeqwristoÔ kl�dou twn majhmatik¸n.

Mia exÐswsh thc morf c

f(x, y, z, . . . ) = 0,

ìpou f(x, y, z, . . . ) eÐnai èna polu¸numo metablht¸n x, y, z, . . . me akèraiouc sun-telestèc, onom�zetai Diofantik exÐswsh kai to prìblhma thc eÔreshc twn akè-raiwn lÔsewn miac tètoiac exÐswshc lègetai Diofantikì prìblhma. 'Otan lème“èstw h Diofantik exÐswsh f(x, y, z, . . . ) = 0” ennooÔme to Diofantikì prì-blhma.

Mia Diofantik exÐswsh mporeÐ na mhn èqei lÔseic, na èqei peperasmènopl joc lÔsewn na èqei �peirec lÔseic. Sthn teleutaÐa perÐptwsh oi lÔseicsun jwc dÐdontai sunart sei miac perissìterwn akèraiwn paramètrwn.

Gewmetrik�, oi akèraiec lÔseic thc Diofantik c exÐswshc f(x, y) = 0 pari-stoÔn ta shmeÐa epÐ thc kampÔlhc f(x, y) = 0 pou èqoun akèraiec suntetagmènec.Gia par�deigma, sthn perÐptwsh thc kampÔlhc x2−2y2 = 0, h mình akèraia lÔsheÐnai profan¸c h (x, y) = (0, 0), dhlad to shmeÐo (0, 0) eÐnai to mìno shmeÐo epÐtwn dÔo eujei¸n x2 − 2y2 = 0 me akèraiec suntetagmènec. Sthn perÐptwsh thckampÔlhc x + 2y − 1 = 0 èqoume tic �peirec lÔseic (x, y) = (3 + 2k,−1 − k)

k ∈ Z, en¸ sthn perÐptwsh thc kampÔlhc 4x+6y = 11 den èqoume kamÐa akeraÐalÔsh (afoÔ gia k�je akèraiouc x kai y to aristerì mèloc eÐnai �rtioc arijmìcen¸ to dexiì eÐnai perittìc arijmìc).

'Estw α1, α2, . . . , αn, βn ∈ Z. H pio apl Diofantik exÐswsh n metablht¸nx1, x2, . . . , xn eÐnai thc morf c

α1x1 + α2x2 + · · ·+ αnxn = βn(*)

Aut onom�zetai grammik Diofantik exÐswsh n metablht¸n. H onomasÐaproèrqetai apì to gegonìc ìti h grammik Diofantik exÐswsh dÔo metablh-t¸n α1x1 + α2x2 = β parist� eujeÐa gramm sto epÐpedo.


Skopìc mac ed¸ eÐnai na kajorÐsoume ìlec tic akèraiec lÔseic thc (∗). Giato lìgo autì qreiazìmaste thn ènnoia twn m.k.d kai e.k.p perissìterwn twn dÔoakèraiwn arijm¸n.

H ènnoia tou m.k.d. dÔo akèraiwn epekteÐnetai kai se perissìterouc twn dÔo.An α1, α2, . . . , αn eÐnai akèraioi, ìqi ìloi Ðsoi me to mhdèn, tìte autoÐ èqoun koi-noÔc diairètec, gia par�deigma touc ±1. To pl joc twn koin¸n diairet¸n aut¸neÐnai peperasmèno. O megalÔteroc twn koin¸n diairet¸n twn α1, α2, . . . , αn, ono-m�zetai mègistoc koinìc diairèthc kai sumbolÐzetai me (α1, α2, . . . , αn). AutìceÐnai ≥ 1. An (α1, . . . , αn) = 1, tìte lème ìti oi α1, α2, . . . , αn eÐnai sqetik� pr¸-toi metaxÔ touc. An (αi, αj) = 1, i, j = 1, 2, . . . , n tìte lème ìti oi α1, α2, . . . , αn

eÐnai an� dÔo sqetik� pr¸toi metaxÔ touc. Profan¸c an oi α1, α2, . . . , αn eÐnaian� dÔo sqetik� pr¸toi tìte eÐnai kai sqetik� pr¸toi metaxÔ touc. Prosoq , toantÐstrofo den isqÔei.

Me an�logo trìpo epekteÐnetai kai h ènnoia tou e.k.p. dÔo akèraiwn seperissìterouc twn dÔo mh mhdenik¸n akèraiwn α1, α2, . . . , an, orÐzontac wc el�-qisto koinì pollapl�siì touc to mikrìtero jetikì akèraio (pou up�rqei lìgwthc arq c tou elaqÐstou) metaxÔ ìlwn twn koin¸n pollaplasÐwn touc (pou eÐnai�peirou pl jouc). Autì sumbolÐzoume me [α1, α2, . . . , αn].

EÔkola prokÔptei ìti isqÔei

(α1, α2, . . . , αn) = ((α1, α2, . . . , αn−1), αn) = ((α1, . . . , αk), . . . , (αm, . . . , αn))

kai

[α1, α2, . . . , αn] = [[α1, α2, . . . , αn−1], αn] = [[α1, . . . , αk], . . . , [αm, . . . , αn]].

Oi idiìthtec de pou isqÔoun gia ton m.k.d. kai to e.k.p. epekteÐnontai kai giaperissìterouc twn dÔo. Gia par�deigma, èqoume ìti

• To e.k.p. [a1, a2, . . . , an] diaireÐ ìla ta koin� pollapl�sia twn a1, a2, . . . , an.

• An δ = (α1, α2, . . . , αn) tìte up�rqoun akèraioi x1, x2, . . . , xn ètsi ¸ste

δ = α1x1 + α2x2 + · · ·+ αnxn

kai k�je diairèthc twn α1, α2, . . . , αn eÐnai diairèthc tou δ.


EpÐshc o δ eÐnai o mikrìteroc ìlwn twn ekfr�sewn α1y1+α2y2+· · ·+αnyn >

0, me y1, . . . , yn ∈ Z, kai m�lista ìlec autèc oi ekfr�seic eÐnai pollapl�sia touδ.

H sqèsh 1.1.8 pou sundèei ton m.k.d. kai to e.k.p. dÔo akèraiwn epekteÐnetaise perissìterouc twn dÔo mh mhdenik¸n jetik¸n akèraiwn α1, α2, . . . , αn wcex c.

[a1, a2, . . . , an] =∏

(αi1 , αi2 , . . . , αim)∏(αj1 , αj2 , . . . , αjm)

ìpou to ginìmeno ston arijmht (ant. sto paronomast ) lamb�netai wc procìlec tic m-idec (i1, i2, . . . , im) (ant. (j1, . . . , jm)) ìpou 1 ≤ i1 < i2 < · · · <

im ≤ n gia m perittì (ant. 1 < j1 < j2 < · · · < jm < n, gia m �rtio). Giapar�deigma èqoume

[α1, α2, α3] =α1α2α3(α1, α2, α3)

(α1, α2)(α1, α3)(α2, α3)

kai

[α1, α2, α3, α4] =α1α2α3α4(α1, α2, α3)(α1, α2, α4)(α1, α3, α4)(α2, α3, α4)

(α1, α2)(α1, α3)(α1, α4)(α2, α3)(α2, α4)(α3, α4)(α1, α2, α3, α4).

Pr�gmati, gr�foume touc αi se ginìmeno pr¸twn:

αi =∏

pvk(αi)k .

Opìte arkeÐ na deÐxoume ìti

max{α1, . . . , αn} =n∑

m=1

(−1)m+1 min{αi1 , αi2 , . . . , αim}.

ParathroÔme ìti aut h sqèsh eÐnai summetrik wc proc ta αi, dhlad an efar-mosjeÐ opoiad pote met�jesh sta α1, α2, . . . , αn h sqèsh aut paramènei h Ðdia.Sunep¸c mporoÔme na upojèsoume ìti α1 ≥ α2 ≥ α3 ≥ · · · ≥ αn. 'Ara aut hsqèsh eÐnai isodÔnamh me thn

α1 =n∑

m=1

(−1)m+1αim


pou profan¸c isqÔei.'Estw t¸ra δn = m.k.d.(α1, α2, . . . , αn) ìpou αi eÐnai oi suntelestèc pou

emfanÐzontai sth Diofantik ExÐswsh (∗). Sta prohgoÔmena eÐdame ìti o δn

eÐnai o mikrìteroc akèraioc tou sunìlou {α1x1+· · ·+αnxn > 0/x1, . . . , xn ∈ Z}.Epiplèon k�je stoiqeÐo tou sunìlou

M = {α1x1 + · · ·+ αnxn/x1, . . . , xn ∈ Z}

eÐnai èna pollapl�sio tou δn, dhlad èqoume

M = δnZ = {δnm/m ∈ Z}.

Sunep¸c èqoume

1.1.22 L mma. H (∗) èqei lÔsh an kai mìnon an o βn diaireÐtai dia tou m.k.d.(α1, α2, . . . , αn).

T¸ra prosdiorÐzoume ìlec tic lÔseic thc (∗).

1.1.23 Je¸rhma. 'Estw α1, α2 dÔo mh mhdenikoÐ akèraioi kai èstw δ2 =

(α1, α2). Tìte h exÐswshα1x1 + α2x2 = β2

èqei lÔsh an kai mìnon an δ2|β2. Epiplèon, an (u1, u2) eÐnai mia lÔsh, tìte k�je�llh lÔsh eÐnai thc morf c

x1 = u1 + kα2

δ2, x2 = u2 − k

α1

δ2.

Apìdeixh. O pr¸toc isqurismìc dÐdetai sto prohgoÔmeno L mma. 'Estw loipìnìti δ2|β2 kai ìti (u1, u2) eÐnai mia lÔsh. Tìte h antikat�stash twn x1 kai x2 sthnexÐswsh me touc akèraiouc u1 + k

α2

δ2kai u2 − k

α1

δ2antÐstoiqa, deÐqnei ìti autoÐ

thn epalhjeÔoun. Autì apodeiknÔetai kai gewmetrik� kaj¸c o −α1

α2= −α1/δ2

α2/δ2eÐnai h klÐsh thc eujeÐac α1x1 + α2x2 = β2 sto epÐpedo.

AntÐstrofa, t¸ra apodeiknÔoume ìti an u1, u2 eÐnai mia lÔsh tìte k�je �llhlÔsh èqei th morf pou anafèretai sto je¸rhma. 'Estw (x1, x2) mia opoiad pote


lÔsh, opìte èqoume

α1x1 + α2x2 = β2 kai α1u1 + α2u2 = β2.

'Araα1(x1 − u1) = α2(u2 − x2)

α1

δ2(x1 − u1) =

α2

δ2(u2 − x2).

All�(

α1

δ2,α2

δ2

)= 1, opìte apì to L mma tou EukleÐdh prèpei

α2

δ2

∣∣∣x1 − u1 kaiα1

δ2

∣∣∣u2 − x2.

'Arax1 − u1 = k

α2

δ2 x1 = u1 + k

α2

δ2,

gia k�poio k ∈ Z. 'Etsi èqoume

α1

δ2

(u1 + k

α2

δ2− u1

)=

α2

δ2(u2 − x2) k

α1

δ2= u2 − x2,

dhlad x2 = u2 − kα1

δ2.

To Ðdio apotèlesma mporeÐ na exaqjeÐ kai qwrÐc th qr sh tou L mmatoc touEukleÐdh wc ex c. Upojètoume ìti (v1, v2) eÐnai mia lÔsh thc

α1x1 + α2x2 = δ2(1)

dhlad

α1v1 + α2v2 = δ2(2)

kai èstw (x1, x2) mia opoiad pte lÔsh thc (1).Pollaplasi�zontac thn (1) kai thn (2) epÐ v1 kai epÐ x1 antÐstoiqa, paÐrnoume

α1x1v1 + α2x2v1 = v1δ2

α1x1v1 + α2x1v2 = x1δ2,

opìteα2(x2v1 − x1v2) = δ2(v1 − x1).


EpÐshc pollaplasi�zontac thn (1) kai thn (2) epÐ tou v2 kai epÐ x2 antÐstoiqa,èqoume

α1v2x1 + α2v2x2 = v2δ2

α1v1x2 + α2v2x2 = x2δ2

opìte

α1(v2x1 − v1x2) = δ2(v2 − x2).

Jètontac x2v1 − x1v2 = k, paÐrnoume

kaiδ2(v1 − x1) = α2k, dhlad x1 = v1 +

α2

δ2k

δ2(v2 − x2) = −α1k, dhlad x2 = v2 − α1

δ2k.

An β2 = β′2δ2, tìte h (u1, u2), u1 = β′2v1, u2 = β′2v2, eÐnai mia lÔsh thc arqik cexÐswshc kai h opoiad pote �llh lÔsh ja eÐnai (β′2x1, β

′2x2), ìpou

β′2x1 = u1 +α2

δ2β′2k = u1 +

α2

δ2k′

k′ ∈ Z.

β′2x2 = u2 − α1

δ2β′2k = u2 − α1

δ2k′

Parat rhsh. Gr�fontac thn exÐswsh α1x1 + α2x2 = β2 sth morf

x2 = −α1

α2x1 +

β2

α2,

blèpoume ìti h klÐsh thc eujeÐac pou parist� aut h exÐswsh eÐnai jetik ìtan taα1 kai α2 eÐnai eterìshma kai sunep¸c èna tm ma thc eujeÐac brÐsketai sto pr¸totetarthmìrio kai perilamb�nei �peiro pl joc shmeÐwn autoÔ tou tetarthmorÐou.An (x1, x2) eÐnai èna shmeÐo autoÔ tou tetarthmorÐou, dhlad x1 > 0, x2 > 0,pou brÐsketai epÐ thc eujeÐac kai eÐnai x1, x2 ∈ Z, dhlad eÐnai mia jetik akèraialÔsh thc exÐswshc, tìte up�rqoun �peiro pl joc jetikèc akèraiec lÔseic thcexÐswshc. An h klÐsh eÐnai arnhtik , dhlad ta α1 kai α2 eÐnai kai ta dÔo jetik�


kai ta dÔo arnhtik�, kai èna tm ma thc eujeÐac brÐsketai sto pr¸to tetarthmìriatìte h exÐswsh èqei to polÔ èna peperasmèno pl joc jetik¸n akèraiwn lÔsewn.Argìtera ja broÔme autì to pl joc.

Par�deigma. 1. Na lujeÐ h Diofantik exÐswsh

392x1 − 21x2 = 14.

BrÐskoume ton m.k.d. (392, 21), me ton EukleÐdeio algìrijmo. 'Eqoume

392 = 21 · 18 + 14

21 = 14 · 1 + 7

14 = 7 · 2 + 0.

'Ara

(392, 21) = 7 = 21− 14 · 1 = 21− (392− 21 · 18) · 1= −392 + 21 · 19 = 392(−1) + 21(19).

Sunep¸c 392(−2)−21(−38) = 14, dhlad mia lÔsh thc dosmènhc exÐswshc eÐnaih (−2,−38). Opìte ìlec oi lÔseic eÐnai oi (−2− 3t,−38− 56t), t ∈ Z. EpÐshcparathroÔme ìti gia t = −1 paÐrnoume th mikrìterh jetik akèraia lÔsh aut cthc exÐswshc, dhlad thn (1, 18). Opìte ìlec oi lÔseic thc exÐswshc mporoÔnna dÐdontai kai wc oi (1− 3k, 18− 56k), k ∈ Z. EpÐshc shmei¸noume ìti kaj¸co k diatrèqei ìlouc touc akèraiouc, o −k p�li diatrèqei ìlouc touc akèraiouc,opìte ìlec oi lÔseic mporoÔn na ekfrasjoÔn kai wc (1 + 3k, 18 + 56k), k ∈ Z.Ap� autì blèpoume �mesa ìti up�rqoun �peirec jetikèc akèraiec lÔseic, dhlad blèpoume autì pou dhl¸netai kai apì th jetik klÐsh thc eujeÐac.

To prohgoÔmeno par�deigma ja mporoÔse na tejeÐ kai wc ex c. Na brejeÐo mikrìteroc jetikìc akèraioc arijmìc o opoioc ìtan diairejeÐ dia tou 392 kaidia tou 21 af nei upìloipo 3 kai 17 antÐstoiqa. Dhlad ja prèpei na isoÔte me392x1 + 3 kai me 21x2 + 17 kai �ra

392x1 − 21x2 = 14.


'Opwc eÐdame o mikrìteroc tètoioc akèraioc eÐnai o

392 · (1) + 3 = 21(18) + 17 = 395.

Par�deigma. 2. Na brejeÐ to pl joc twn jetik¸n akèraiwn lÔsewn thcDiofantik c exÐswshc

392x1 + 21x2 = 14.

'Opwc eÐdame eÐnai (392, 21) = 7 = 392(−1) + 21(19). Sunep¸c 392(−2) +

21(38) = 14 kai h genik lÔsh thc exÐswshc t¸ra eÐnai h (−2 + 3t, 38 − 56t),t ∈ Z. Oi jetikèc lÔseic eÐnai autèc gia tic opoÐec t >

32

kai t <3856

, dhlad 32

< t <3856

pou den mporeÐ na isqÔei, dhlad den up�rqei tètoio t kai �ra denup�rqei kami� jetik akèraia lÔsh, ìpwc �llwste �mesa faÐnetai gewmetrik�sto sq ma:

CCCCCCCCCCCCCCCCCC

x2

•23

•256

x1

1.1.24 Je¸rhma. H exÐswsh (∗) èqei lÔsh an kai mìnon an δn|βn. Epiplèonan δi eÐnai o m.k.d. (α1, α2, . . . , αi) kai x

(i)1 , x

(i)2 , . . . , x

(i)i ) eÐnai mia lÔsh thc

δi = α1x1 + α2x2 + · · ·+ αixi, i = 2, . . . , n tìte ìlec oi lÔseic thc (∗) eÐnai thc


morf c

x1

x2

x3

x4

...xn−1

xn

=

0BBBBBBBBBBBBBBBBBBBB@

α2δ2

α3δ3

x(2)1

α4δ4

x(3)1 ··· αn−1

δn−1x(n−2)1

αnδn

x(n−1)1 x

(n)1

−α1δ2

α3δ3

x(2)2

α4δ4

x(3)2 ··· αn−1

δn−1x(n−2)2

αnδn

x(n−1)2 x

(n)2

0 − δ2δ3

α4δ4

x(3)3 ··· αn−1

δn−1x(n−2)3

αnδn

x(n−1)3 x

(n)3

0 0 − δ3δ4

··· αn−1

δn−1x(n−2)4

αnδn

x(n−1)4 x

(n)4

......

... ···...

......

0 0 0 ··· − δn−2

δn−1

αnδn

x(n−1)n−1 x

(n)n−1

0 0 0 ··· 0 − δn−1

δn−1x(n)n

1CCCCCCCCCCCCCCCCCCCCA

0BBBBBBBBBBBBBBBBBBBB@

t1

t2

t3

t4

...tn−1

tn

1CCCCCCCCCCCCCCCCCCCCAìpou βn = tnδn kai ti ∈ Z, i = 1, 2, . . . , n− 1.

Apìdeixh. To pr¸to mèroc tou jewr matoc eÐnai to 1.1.22. Gia to deÔtero mèroctou jewr matoc efarmìzoume epagwg sto n. H perÐptwsh n = 2 dÐdetai stoJe¸rhma 1.1.23. Upojètoume ìti to je¸rhma isqÔei gia n = k ≥ 2 kai èstwx

(k+1)1 , x

(k+1)2 , . . . , x

(k+1)k+1 eÐnai mia lÔsh thc δk+1 = α1x1 + · · · + αk+1xk+1 kai

ìti x1, . . . , xk+1 eÐnai mia lÔsh thc tk+1δk+1 = α1x1 + · · ·αk+1xk+1, ìpou tk+1,α1, α2, . . . , αk+1 eÐnai mh mhdenikoÐ akèraioi. Opìte èqoume

α1(x1 − tk+1x(k+1)1 ) + · · ·+ αk+1(xk+1 − tk+1x

(k+1)k+1 ) = 0

α1

δk+1(x1−tk+1x

(k+1)1 )+· · ·+ αk

δk+1(xk−tk+1x

(k+1)k ) = −αk+1

δk+1(xk+1−tk+1x

(k+1)k+1 ).

T¸ra to �jroisma α1(x1 − tk+1x(k+1)1 ) + · · ·αk(xk − tk+1x

(k+1)k ) eÐnai èna pol-

lapl�sio tou m.k.d. δk twn α1, α2, . . . , αk, èstw λδk. Opìte to aristerì mèlocthc prohgoÔmenhc isìthtac eÐnai λδk/δk+1.

All� oi akèraioiα1

δk+1,

α2

δk+1, . . . ,

αk

δk+1,αk+1

δk+1eÐnai sqetik� pr¸toi metaxÔ

touc kai sunep¸c(

δk

δk+1,αk+1

δk+1

)= 1. 'Ara

αk+1

δk+1tk = λ. Opìte h teleutaÐa

isìthta isqÔei an kai mìnon an λδk/δκ+1 =αk+1

δk+1tk+1

δk

δk+1= −αk+1

δk+1(xk −

tk+1x(k+1)k+1 ) xk+1 − tk+1x

(k+1)k+1 = −tk

δk

δk+1kai

α1

δk+1(x1 − tk+1x

(k+1)1 ) + · · ·+

αk

δk+1(xk − tk+1x

(k+1)k+1 ) = tk

αk+1

δk+1δk.


Apì thn upìjesh thc epagwg c, gia k�je eklog tou tk, èqoume

x1 − tk+1x(k+1)1 = tk

αk+1

δk+1x

(k)1 + tk−1

αk

/δk+1

δk

/δk+1

x(k−1)1 + · · ·+ t1

α2

/δk+1

δ2

/δk+1

x2 − tk+1x(k+1)2 = tk

αk+1

δk+1x

(k)2 + tk−1

αk

/δk+1

δk

/δk+1

x(k−1)2 + · · · − t1

α1

/δk+1

δ2

/δk+1

............................................................................................................

xk − tk+1x(k+1)k = tk

αk+1

δk+1x

(k)k − tk−1

δk−1

/δk+1

δk

/δk+1

,

ìpou t1, t2, . . . , tk−1 eÐnai k�poioi akèraioi kaiα1

δk+1x

(i)1 + · · ·+ αk

δk+1x

(i)k =

δi

δk+1gia i = 2, . . . , k.

Metafèrontac touc ìrouc pou perilamb�noun ta tk+1 sto dexiì mèloc twnprohgoÔmenwn isot twn kai aplopoi¸ntac touc upìloipouc ìrouc paÐrnoume toepijumhtì apotèlesma.

Sunep¸c an broÔme mia lÔsh thc (∗) tìte gnwrÐzoume ìlec tic lÔseic. Giathn eÔresh miac lÔshc thc (∗) ergazìmaste wc ex c: JewroÔme thn exÐswsh

α1x1 + α2x2 + · · ·+ αnxn = δn(**)

ìpou ìloi oi suntelestèc αi eÐnai 6= 0 (diaforetik� ja eÐqame mia exÐswsh m(<

n) metablht¸n). MporoÔme na upojèsoume ìti ìloi oi suntelestèc αi eÐnai > 0,diìti diaforetik� mporoÔme na antikatast soume k�je arnhtikì αi me ton −αi

(pou den all�zei thn tim tou m.k.d. δn) kai to antÐstoiqo xi me to −xi.GnwrÐzoume ìti δn = (α1, α2, . . . , αn) = ((α1, . . . , αn−1), , αn) = (δn−1, δn).

Sunep¸c an gnwrÐzoume mia lÔsh (x(n−1)1 , . . . , x

(n−1)n−1 ) thc

α1x1 + · · ·+ αn−1xn−1 = δn−1

tìte brÐskoume (me ton algìrijmo tou EukleÐdh me �llo trìpo) mia lÔsh thcδn−1x1 + αnxn = δn, èstw thn (y(n)

1 , x(n)n ), opìte h (x(n)

1 , x(n)2 , . . . , x

(n)n ), ìpou

x(n)1 = x

(n−1)1 y

(n)1 , . . . , x

(n)n−1 = x

(n−1)n−1 y

(n)1 , eÐnai lÔsh thc (∗∗). 'Ara epagwgik�,

xekin¸ntac apì thn eÔresh miac lÔshc thc α1x1+α2x2 = δ2, brÐskoume mia lÔshthc α1x1 + α2x2 + α3x3 = δ3, katìpin thc α1x1 + α2x2 + α3x3 + α4x4 = δ4


k.o.k. èwc ìtou na broÔme mia lÔsh thc (∗∗). An βn = tnδn, tìte mia lÔsh thc(∗) eÐnai h (tnx

(n)1 , . . . , tnx

(n)n ).

Par�deigma. Na lujeÐ h Diofantik exÐswsh

5x1 + 35x2 + 8x3 + 7x4 = 2.

'Eqoume δ4 = (5, 35, 8, 7) = 1. 'Ara aut èqei lÔsh.JewroÔme thn

5x1 + 35x2 + 8x3 + 7x4 = 1.

BrÐskoume mia lÔsh thc 5x1 + 35x2 = 5 thc x1 + 7x2 = 1 pou profan¸c mialÔsh thc eÐnai h (8,−1).

Katìpin brÐskoume mia lÔsh thc 5q1 + 8q2 = 1 pou profan¸c mia lÔsh thceÐnai h (−3, 2), opìte h (−24, 3, 2) eÐnai mia lÔsh thc 5x1 + 35x2 + 8x3 = 1.

Tèloc jewroÔme thn exÐswsh x1 + 7x4 = 1. Mia lÔsh aut c eÐnai h (−6, 1)

kai �ra h (2 · 144,−2 · 18,−2 · 12, 2 · 1) eÐnai lÔsh thc arqik c exÐswshc.Sunep¸c, sÔmfwna me to 1.1.24, h genik thc lÔshc (x1, x2, x3, x4) eÐnai h

x1

x2

x3

x4

=

7 64 −168 144

−1 −8 21 −18

0 −5 14 −12

0 0 −1 1

t1

t2

t3

2

Mèjodoc Weinstock. 'Eqoume dei ìti o m.k.d. δn mh mhdenik¸n akeraÐwnα1, α2, . . . , αn eÐnai o mikrìteroc jetikìc akèraioc thc morf c

α1x1 + α2x2 + · · ·+ αnxn, x1, x2, . . . , xn ∈ Z.

Qrhsimopoi¸ntac autì to gegonìc o R. Weinstock (1960) prìteine ton ex calgìrijmo gia thn eÔresh tou δ all� kai thn eÔresh akeraÐwn x1, . . . , xn ètsi¸ste δ = α1x1 + · · ·+ αnxn.

'Estw β(0)1 , β

(0)2 , . . . , β

(0)n tuqaÐoi akèraioi tètoioi ¸ste

α1β(0)1 + α2β

(0)2 + · · ·+ αnβ(0)

n = γ > 0.


An γ|α1, γ|α2, . . . , γ|αn, tìte γ = δ, (afoÔ δ ≤ γ kai γ|δ, dhlad γ ≤ δ), opìteèqoume brei ton δ. An ìqi, dhlad an γ - αi gia k�poio i, 1 ≤ i ≤ n, tìteαi = qiγ + v1, 0 < v1 < γ. 'Etsi èqoume

ìpouv1 = αi − q1γ = α1β

(1)1 + α2β

(1)2 + · · ·+ αnβ(1)

n

β(1)j = −q1β

(0)j , j 6= i, kai β

(1)i = 1− q1β

(0)i .

Epanalamb�nontac aut th diadikasÐa paÐrnoume mia austhr� fjÐnousa akolou-jÐa {vi} akèraiwn arijm¸n thc morf c

α1x1 + α2x2 + · · ·+ αnxn ≥ δ

kai �ra met� apì èna peperasmèno pl joc bhm�twn aut c thc diadikasÐac ja fj�-soume se ènan deÐkth k gia ton opoÐon gia pr¸th for� ja èqoume vk|α1, vk|α2, . . . , vk|αn,opìte δ = vk = α1β

(k)1 + α2β

(k)2 + · · ·+ αnβ

(k)n . Dhlad h (β(k)

1 , β(k)2 , . . . , β

(k)n )

eÐnai mia lÔsh thc δ = α1x1 + · · ·+ αnxn.

Par�deigma. Ac jewr soume p�li ton m.k.d. δ = (5, 35, 8, 7). DialègoumetuqaÐouc β

(0)1 , β

(0)2 , β

(0)3 , β

(0)4 ètsi ¸ste

γ = 5β(0)1 + 35β

(0)2 + 8β

(0)3 + 7β

(0)4 > 0.

FerepeÐn, β(0)1 = 0, β

(0)2 = 0, β

(0)3 = 1, β

(0)4 = 0. EÐnai γ = 7 kai epeid γ - 5,

prèpei γ > δ. DiairoÔme to 5 dia 7 : 5 = 7 · 0 + 5, opìte

0 < 5 = 5− 7 · 0 = 5 · 1 + 35 · 0 + 8 · 0 + 7 · 0 < 7.

Epeid 5 - 7, prèpei 5 > δ kai èqoume 7 = 5 · 1 + 2. 'Ara 2 = 7 − 5 · 1 =

5(−1) + 35 · 0 + 8 · 0 + 7(1 − 0). P�li epeid 2 - 5, prèpei 2 > δ kai èqoume5 = 2 · 2 + 1. Opìte 1 = 5(1− 2 · (−1)) + 35 · 0 + 8 · 0 + 7(−2).

T¸ra profan¸c δ = 1 kai h (β(3)1 = 3, β

(3)2 = 0, β

(3)3 = 0, β

(3)4 = −2) eÐnai

mia lÔsh thc 1 = 5x1 + 35x2 + 8x3 + 7x4.

Mèjodoc Euler. 'Estw h exÐswsh

51x1 + 31x2 + 14x3 = 1.


O Leonard Euler sto biblÐo tou “Vollstandige Anleitung zur Algebra. St.

Petersburg 1770” gia na brei tic akèraiec lÔseic grammik¸n Diofantik¸n exis¸-sewn akoloujeÐ thn ex c diadikasÐa:

Dialègoume ton mikrìtero kat� apìluth tim suntelest thc exÐswshc dhla-d to 14 (An up�rqoun �lloi suntelestèc sthn exÐswsh pou eÐnai pollapl�siaautoÔ tìte ant� autoÔ jewroÔme to megalÔtero aut¸n). Gr�foume

14x3 = −51x1 − 31x2 + 1 x3 = −3x1 − 2x2 +−9x1 − 3x2 + 1

14.

Epeid endiaferìmeja gia tic akèraiec lÔseic, an (x1, x2, x3) eÐnai mia tètoia,tìte ja prèpei o arijmìc

−9x1 − 3x2 + 114

na eÐnai akèraioc. Jètoume

t =−9x1 − 3x2 + 1

14.

Opìte èqoume 9x1 + 3x2 + 14t = 1. Epeid o 3 eÐnai o mikrìteroc suntelest cmetaxÔ twn x1 kai x2 all� o suntelest c 9 eÐnai pollapl�siìc tou, jewroÔmeton suntelest 9 kai epanalamb�noume thn Ðdia diadikasÐa ìpwc prin. 'Eqoume

x1 = −t +−5t− 3x2 + 1

9

ìpou t¸ra prèpei o u =−5t− 3x2 + 1

9na eÐnai akèraioc. 'Etsi èqoume

3x2 + 5t + 9u = 1

opìte

x2 = −t− 3u +−2t + 1

3.

Jètoume v =−2t + 1

3(ton opoÐo jewroÔme akèraio) kai èqoume 3v = −2t + 1

t = −v +−v + 1

2me ω =

−v + 12

∈ Z, 2ω = −v + 1 v = 1− 2ω.

Antikajist¸ntac to v sto t kai autì sto x2, èqoume t = −1 + 3ω kaix2 = −(−1 + 3ω)− 3u + 1− 2ω, opìte x2 = 2− 5ω − 3u. EpÐshc eÐnai

x1 = 1− 3ω + u


kai

x3 = −3(1− 3ω + u)− 2(2− 5ω − 3u)− 1 + 3ω = −8 + 22ω + 3u.

Opìte h genik lÔsh dÐnetai wc h

x1 = u− 3ω + 1

x2 = −3u− 5ω + 2

x3 = 3u + 22ω − 8.

An jewr soume th lÔsh (1, 2,−8), tìte apì to Je¸rhma 1.1.24 h genik lÔshdÐdetai wc h

x1

x2

x3

=

31 196 1

−51 −322 2

0 −1 −8

t1

t2

1

afoÔ 51(14)+31(−23) = 1, dhlad x(2)1 = 14 kai x(2)

2 = −23, x(3)1 = 1, x

(3)2 = 2

kai x(3)3 = −8. 'Ara ja prèpei na isqÔei

kai31t1 + 196t2 = u− 3ω

−51t1 − 322t2 = −3u− 5ω

kai −t2 = 3u + 22ω

gia t1, t2, u, ω ∈ Z. Pr�gmati, blèpoume ìti gia t2 = −3u + 22ω ∈ Z èqoume

−51t1 + 322(3u + 22ω) = −3u− 5ω

−51t1 = −969u− 7089ω

t1 = 19u + 139ω ∈ Z.

EpÐshc to Ðdio prokÔptei apì thn pr¸th isìthta, afoÔ

31t1 = 589u + 4309ω

t1 = 19u + 139ω.


Parat rhsh. H orÐzousa |A| tou n× n pÐnaka A sto Je¸rhma 1.1.24 isoÔ-tai me 1. Pr�gmati an pollaplasi�soume thn i-gramm epÐ αi gia k�je i =

1, 2, . . . , n, tìte h orÐzousa tou pÐnaka pou ja prokÔyei isoÔte me α1α2 · · ·αn|A|.All� h orÐzousa tou pÐnaka autoÔ eÐnai Ðsh me thn orÐzousa tou pÐnaka pou pro-kÔptei an sthn i-gramm prosjèsoume ìlec tic j-grammèc gia j < i. O pÐnakacautìc eÐnai �nw trigwnikìc me diag¸nio Ðsh me

(α1α2

δ2,δ2α3

δ3,δ3α4

δ4, . . . ,

δn−1αn

δn, δn

).

All� h orÐzousa autoÔ tou pÐnaka eÐnai Ðsh me

α1α2

δ2· δ2α3

δ3· · · δn−1αn

δn, δn = α1α2α3 · · ·αn.

Sunep¸c èqoume α1α2 · · ·αn|A| = α1α2 · · ·αn. Opìte |A| = 1. ShmeÐwse ìti

t1

t2...1

= A−1

x1

x2

...xn

.

Gia par�deigma, prohgoumènwc eÐqame A =

31 196 1

−51 −322 2

0 −1 −8

opìte

t1

t2

1

=

2578 1567 714

−408 −248 −113

−51 −31 14

x1

x2

x3

.

H Katanom twn Pr¸twn

S mera, me th bo jeia isqur¸n hlektronik¸n upologist¸n, èqoun katagrafeÐmakroskeleÐc kat�logoi pr¸twn arijm¸n (blèpe google: The list of prime num-

bers).


Melet¸ntac ènan tètoio kat�logo, to pr¸to qarakthristikì pou diakrÐnoumesthn akoloujÐa

2 = p1 < 3 = p2 < p3 < · · ·

twn pr¸twn arijm¸n eÐnai h apousÐa enìc kanìna o opoÐoc kajorÐzei ton epìme-no pr¸to arijmì apì ènac dosmèno pr¸to arijmì. 'Oqi mìno den mporoÔme nadiakrÐnoume ènan tètoio kanìna all� oÔte kan na mantèyoume k�poia eikasÐa giaton trìpo pou oi pr¸toi arijmoÐ ekteÐnontai kat� m koc twn fusik¸n arijm¸n.Exet�zontac arket� meg�louc arijmoÔc blèpoume ìti oi pr¸toi arijmoÐ spanÐ-zoun all� tautìqrona mporeÐ na emfanÐzontai kont� o ènac me ton �llo. Giapar�deigma, stouc 100 arijmoÔc amèswc prin to 10.000.000, dhlad metaxÔ tou9.999.900 kai 10.000.000, up�rqoun ennèa pr¸toi pou eÐnai oi

9.999.901, 9.999.907, 9.999.929, 9.999.931, 9.999.937

9.999.943, 9.999.971, 9.999.973, kai 9.999.991

en¸ up�rqoun mìno dÔo pr¸toi stouc epìmenouc ekatì arijmoÔc apì ton 10.000.000èwc ton 10.000.100 pou eÐnai oi 10.000.019 kai 10.000.079. EpÐshc den up�rqeikanènac pr¸toc metaxÔ twn arijm¸n 20.831.323 kai 20.831.533 all� oi arijmoÐ1.000.000.000.061 kai 1.000.000.000.063 eÐnai pr¸toi. Aut� ta arijmhtik� apo-telèsmata dikaiologoÔntai kai apì to gegonìc ìti gia k�je fusikì arijmì n oidiadoqikoÐ arijmoÐ

(n + 1)! + 2, (n + 1)! + 3, . . . , (n + 1)! + (n + 1)

eÐnai ìloi sÔnjetoi. Autì shmaÐnei ìti h akoloujÐa twn pr¸twn arijm¸n perièqeidiadoqikoÔc pr¸touc pou h diafor� touc eÐnai polÔ meg�lh. EpÐshc isqÔei h“Arq tou Bertrand” pou anafèrei ìti gia k�je fusikì arijmì n ≥ 2 up�rqeitoul�qiston ènac pr¸toc p me n ≤ p ≤ 2n (opìte pn+1 ≤ 2pn). (H “arq ”aut eÐnai je¸rhma kaj¸c apedeÐqjh apì ton Tshebycheff to 1952. O Bertrand

upojètei aut thn arq gia na deÐxei ìti to pl joc twn pr¸twn eÐnai �peiro wcex c: De mporeÐ na up�rqei arijmìc p pou eÐnai o megalÔteroc pr¸toc, afoÔmetaxÔ tou p + 1 kai tou 2(p + 1) ja up�qei �lloc pr¸toc megalÔteroc tou p).


Oi prohgoÔmenec parathr seic kai epishm�nseic deÐqnoun ìti, exet�zontac“atomik�”, dhlad autoÔc kaj� eautoÔc touc pr¸touc toul�qiston an� mikr�Ðsa diast mata, p.q. an� 100 pr¸touc, eÐnai �gnwstoc o trìpoc pou kajorÐzetaih jèsh touc metaxÔ twn fusik¸n arijm¸n. 'Omwc, ìpwc ja doÔme amèswc t¸ra,an exet�soume touc pr¸touc an� meg�la diast mata kai ìqi Ðsou m kouc all�na aux�noun kat� èna pollapl�sio tou 10 tìte o trìpoc me ton opoÐo oi pr¸-toi ekteÐnontai kat� m koc twn fusik¸n arijm¸n eÐnai arket� omalìc. Gia naperigr�youme autì ton trìpo orÐzoume th sun�rthsh π(x) thc aparÐjmhshc ( katanom c) twn pr¸twn stouc fusikoÔc wc ex c. Gia k�je pragmatikì arijmìx, h tim thc π(x) orÐzetai na eÐnai to pl joc twn pr¸twn arijm¸n pou ≤ x,dhlad

π(x) =∑

p≤x

1.

Gia par�deigma π(1) = 0, π(5) = 3, π(19) = π(22) = 8, π(10, 9) = 4 k.o.k. Hπ(x) profan¸c eÐnai stajer� metaxÔ dÔo pr¸twn (dec grafik par�stash) kaiapì to Je¸rhma tou EukleÐdh èqoume

limx→∞π(x) = ∞.

x-q1

a2

a3

q4

a5

q6

a7

q8

q9

q10

a11

q12

a13

q14

q15

q16

a17

q18

a19

q20

q21

q22

a23

q24

π(x)6

q1

q2

q3

q4

q5

q6

q7

q8

q9

(Grafik par�stash thc π(x). Se k�je pr¸th tim tou x, h π(x) “phd�” kat�mÐa mon�da sto k�jeto �xona).

O pr¸toc pou epinìhse mia mèjodo upologismoÔ thc π(x), onomazìmenh“kìskino”, tan o Eratosjènhc (276�194 p.Q.). Aut sthrÐzetai sto ex c ge-gonìc:


“Up�rqei toul�qiston ènac pr¸toc par�gontac enìc sÔnjetou arijmoÔ n

pou eÐnai ≤ √n”.

Pr�gmati, an n = n1n2 me n1, n2 6= n, èstw n1 ≤ n2, tìte n21 ≤ n1n2 = n

kai �ra n1 ≤√

n. Epeid n1 > 1, o n1 èqei ènan pr¸to par�gontai p kai sunep¸cp ≤ n1 ≤

√n me p|n.

To “koskÐnisma” twn fusik¸n arijm¸n apì to 2 èwc ènan dosmèno arijmìx me th mèjodo tou Eratosjènh eÐnai to ex c: Gr�foume ìlouc touc fusikoÔcarijmoÔc, me th fusik touc di�taxh, apì to 2 èwc to megalÔtero fusikì n = [x]

pou eÐnai mikrìteroc apì ton x (dhlad to akèraio mèroc tou x). Diagr�foumeìla ta pollapl�sia tou 2 ektìc apì to 2 pou eÐnai pr¸toc, dhlad diagr�foumetouc arijmoÔc 2 · 2, 2 · 3, . . . , 2k, . . . gia ìla ta k ≤ 1

2n. O amèswc epìmenoc

met� ton 2 arijmìc pou den èqei diagrafeÐ prèpei na eÐnai pr¸toc (giatÐ;) kaieÐnai o 3. Katìpin diagr�foume ìla ta pollapl�sia tou 3, dhlad touc 3 ·3, 3 · 4, . . . , 3k, . . . gia ìla ta k ≤ 1

3n. O amèswc epìmenoc met� ton 3 pou den

èqei diagrafeÐ prèpei na eÐnai o 5 pou eÐnai pr¸toc afoÔ de diaireÐtai dia tou2 kai tou 3. To epìmeno b ma eÐnai h diagraf ìlwn twn pollaplasÐwn tou 5arqÐzontac apì ton 52, dhlad diagr�foume touc 5·5, 5·6, . . . k.t.l. GnwrÐzontacìti ènac apì touc pr¸touc par�gontec enìc sÔnjetou arijmoÔ prèpei na eÐnaimikrìteroc Ðsoc apì thn tetragwnik rÐza tou arijmoÔ, h diadikasÐa diagraf cja suneqisjeÐ èwc ìtou brejeÐ o megalÔteroc pr¸toc pou eÐnai mikrìteroc apìthn

√n. Shmei¸noume ìti sth diadikasÐa aut ìtan èqoume brei èna pr¸to p

kai diagr�youme ta pollapl�si� tou o amèswc epìmenoc fusikìc pou paramèneieÐnai epÐshc ènac pr¸toc p′ diìti diaforetik� ja èprepe na diaireÐtai apì k�poiopr¸to mikrìtero apì ton Ðdio (m�lista mikrìtero apì thn

√p′) kai ja èprepe

na eÐqe diagrafeÐ. EpÐshc parathroÔme ìti de qrei�zetai na l�boume upìyin tapollapl�sia tou p′ pou eÐnai mikrìtera tou p′2, afoÔ oi sÔnjetoi arijmoÐ oimikrìteroi tou p′2 èqoun ènan pr¸to par�gonta mikrìtero apì ton p′ kai �raèqoun dh diagrafeÐ.

Gia par�deigma ac broÔme ìlouc touc pr¸touc metaxÔ tou 1 kai 150. Diat�-soume autoÔc ìpwc pio k�tw


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´

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´

´

´

´

´

´1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130

131 132 133 134 135 136 137 138 139 140

141 142 143 144 145 146 147 148 149 150

Efarmìzontac to kìskino tou Eratosjènh arkeÐ na broÔme mìno touc pr¸-touc pou eÐnai mikrìteroi tou

√150.

Diagr�foume touc 2 · 2, 2 · 3, . . . , 2 · 75 kai krat�me ton 3.Diagr�foume touc 3 · 3, 3 · 4, . . . , 3 · 50 kai krat�me ton 5.Diagr�foume touc 5 · 5, 5 · 6, . . . , 5 · 30 kai krat�me ton 7.Diagr�foume touc 7 · 7, 7 · 8, . . . , 7 · 21 kai krat�me ton 11.Diagr�foume touc 11 · 11, 11 · 12, 11 · 13.

ParathroÔme ìti 13 >√

150 kai �ra prèpei ìloi oi paramènontec fusikoÐarijmoÐ, dhlad autoÐ pou den èqoun diagrafeÐ eÐnai ìloi oi pr¸toi metaxÔ tou 1kai 150, pou eÐnai oi: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149.

Sto prohgoÔmeno par�deigma eÐqame pènte pr¸touc mikrìterouc thc√

150,dhlad touc 2, 3, 5, 7 kai 11. An diagr�youme kai autoÔc tìte ja parameÐnounoi arijmoÐ 1 kai oi pr¸toi metaxÔ thc

√150 kai tou 150. 'Ara ja p�roume

π(150)− π(√

150) + 1


arijmoÔc. T¸ra parathroÔme ìti sto pr¸to b ma diagraf c apì touc 150 arij-moÔc diagr�foume

[1502

]ap� autoÔc. Sto deÔtero b ma diagr�foume

[1503

]

arijmoÔc all� merikoÐ ap� autoÔc eÐqan dh diagrafeÐ sto pr¸to b ma, afoÔ apìtouc 150 arijmoÔc

[ 1502 · 3

]arijmoÐ diairoÔntai dia tou 2 kai dia tou 3. Sunep¸c

met� to deÔtero b ma diagraf c to pl joc twn arijm¸n pou paramènoun isoÔtaime

150−[1502

]−

[1503

]+

[1502 · 3

].

Sto trÐto b ma to pl joc twn arijm¸n pou diagr�fetai isoÔtai me[1502

]−

[1502 · 5

]−

[1503 · 5

]+

[150

2 · 3 · 5]

.

Pr�gmati, apì touc diagrafìmenouc[150

5

]arijmoÔc èqoun dh diagrafeÐ

[ 1502 · 5

]

kai[ 1503 · 5

]sto pr¸to kai deÔtero b ma antÐstoiqa, all� kai stouc dÔo autoÔc

arijmoÔc perilamb�nontai kai oi arijmoÐ pou diairoÔntai dia 2 ·3 ·5 pou se pl joceÐnai

[ 1502 · 3 · 5

]. Gia to tètarto kai to pèmpto b ma to pl joc twn arijm¸n pou

diagr�fontai isoÔtai antÐstoiqa me[1507

]−

[1502 · 7

]−

[1503 · 7

]−

[1505 · 7

]+

[150

2 · 3 · 7]

+[

1502 · 5 · 7

]+

[150

3 · 5 · 7]

−[

1502 · 3 · 5 · 7

]kai

[15011

]−

[150

2 · 11

]−

[150

3 · 11

]−

[150

5 · 11

]−

[150

7 · 11

]+

[150

2 · 3 · 11

]+

[150

2 · 5 · 11

]

+[

1502 · 7 · 11

]+

[150

3 · 5 · 11

]+

[150

3 · 7 · 11

]−

[150

2 · 3 · 5 · 11

]−

[150

2 · 3 · 7 · 11

]

−[

1502 · 5 · 7 · 11

]−

[150

3 · 5 · 7 · 11

]+

[150

2 · 3 · 5 · 7 · 11

].

Sunep¸c telik� ja èqoume

π(150)− π(√

150) + 1 = 150−N1 −N2 −N3 −N4 −N5

ìpou Ni, i = 1, 2, 3, 4, 5, eÐnai to pl joc twn diagrafèntwn arijm¸n sto i-b ma.


Dhlad , an S = {2, 3, 5, 7, 11}, tìte

π(150)− π(√

150) + 1 = 150−∑

p∈S

[150p

]+

∑p1,p2∈Sp1 6=p2

[150p1p2

]

−∑

p1,p2,p3∈Sp1 6=p2 6=p3 6=p1

[150

p1p2p3

]+

∑p1,p2,p3,p4∈Sp1 6=p2 6=p3 6=p4

p1 6=p4 6=p2,p3 6=p1

[150

p1p2p3p4

]−

[150

2 · 3 · 5 · 7 · 11

]

= 150− (75 + 50 + 30 + 21 + 13) + (25 + 15 + 10 + 6 + 10 + 7 + 4 + 4 + 2 + 1)

− (5 + 3 + 2 + 2 + 1 + 0 + 0) + (0 + 0 + 0 + 0 + 0)− 0.

'Ara π(150) = 35.

Autì to par�deigma mac upodeiknÔei ìti an efarmìsoume epagwg ( efarmì-zontac thn arq tou egklismoÔ-apoklismoÔ) sto pl joc twn pr¸twn pou eÐnaimikrìteroi thc

√x eÔkola prokÔptei ìti isqÔei o tÔpoc:

π(x) = π(√

x)− 1 + [x] +s∑

t=1

(−1)t∑

pi1,...,pit

∈S

diakekrimmèna

[x

pi1 · · · pit

]

ìpou S eÐnai to sÔnolo ìlwn twn pr¸twn p1, p2, . . . , ps pou eÐnai ≤ √x.

O tÔpoc autìc (pou anafèretai wc tÔpoc tou Legendre) eÐnai mèqri s merao mìnoc gnwstìc tÔpoc pou dÐnei tic akribeÐc timèc thc sun�rthshc π(x). All�praktik� autìc o tÔpoc mporeÐ na qrhsimopoihjeÐ gia mikrèc timèc tou x afoÔaux�nontac to x to pl joc twn pr�xewn aux�netai ekjetik�. K�nontac orismènectropopoi seic, gia th beltÐwsh autoÔ tou tÔpou, o Meissel (1870) katìrjwsena upologÐsei thn tim

π(108) = 5.761.455

kai to 1893 o Bertelsen br ke ìti

π(109) = 50.847.478.

Me th bo jeia twn hlektronik¸n upologist¸n èwc s mera èqoun upologisjeÐ


oi timèc π(10k), k ≤ 23 (dec pÐnaka)

x π(n) x/ log x

10 4 4,3100 25 21,7500 95 80,4

JewroÔme t¸ra to kl�smaπ(x)

x. Autì orÐzei to posostì pou analogeÐ stouc

pr¸touc apì ìlouc touc arijmoÔc apì to 1 èwc to x. Gia par�deigma, an x = 10

tìte π(x) = 4 kai sunep¸c ta25

twn 10 arijm¸n eÐnai pr¸toi, dhlad to 40 thcekatì twn arijm¸n apì to 1 èwc to 10 eÐnai pr¸toi. An x = 150 tìte π(x) = 35,opìte to 23,333 thc ekatì twn arijm¸n apì to 1 èwc to 150 eÐnai pr¸toi. ('AllecanalogÐec anafèrontai ston pÐnaka;).

An jèloume na kajorÐsoume poio eÐnai to posostì twn pr¸twn arijm¸n pouanalogeÐ se ìlouc tou arijmoÔc tìte ja prèpei na exet�soume tic timèc twn kla-sm�twn π(x)/x gia opoiad pote polÔ meg�la x. Sth gl¸ssa twn majhmatik¸nautì kajorÐzetai apì to ìrio

limx→∞

π(x)x

.

Apì touc pÐnakec twn pr¸twn mac dhmiourgeÐtai h aÐsjhsh ìti oi pr¸toi arijmoÐeÐnai polÔ lÐgoi se sqèsh me ìlouc touc arijmoÔc, dhlad eÐnai polÔ ligìteroi apìtouc sÔnjetouc. Sunep¸c gia na epalhjeujeÐ aut h aÐsjhsh prèpei na deiqjeÐìti megal¸nontac to x ta kl�smata π(x)/x suneq¸c mikraÐnoun teÐnontac stomhdèn. Pr�gmati, isqÔei ìti

Je¸rhma. limx→∞

π(x)x

= 0.

Apìdeixh. ArkeÐ na deÐxoume ìti gia k�je ε > 0 up�rqei δ = δ(ε) > 0 pouexart�tai apì to ε ètsi ¸ste π(x)/x < ε gia k�je x > δ, afoÔ π(x)/x ≥ 0

gia ìla ta x > 0. Gi� autì ja qrhsimopoi soume thn “Arq tou Bertrand” pousÔmfwna m� aut an n eÐnai ènac fusikìc arijmìc tìte up�rqei ènac pr¸toc p

tètoioc ¸ste2n−1 < p ≤ 2n.


'Ara p|(2n)! all� p - (2n−1)!. Sunep¸c p∣∣(22

2n−1

), afoÔ (2n)! =

(2n

2n−1

)(2n−1)!2.

EpÐshc efarmìzontac epagwg , lìgw thc sqèshc(m+1

k

)=

(mk

)+

(m

k−1

), genik�

isqÔei 2m ≥ (mk

). Gia m = 2n, èqoume

22n ≥(

2n

2n−1

)≥

∏

2n−1<p≤2n

≥ (2n−1)π(2n)−π(2n−1).

Opìte

2n ≥ (n− 1)(π(2n)− π(2n−1))

π(2n)− π(2n−1) ≤ 2n

n− 1.

Prosjètontac tic anisìthtec

π(23)− π(22) ≤ 23

3− 1

π(24)− π(23) ≤ 24

4− 1...

π(22k)− π(22k−1) ≤ 22k

2k − 1

paÐrnoume

π(22k)− π(22) ≤2k∑

i=3

2i

i− 1

π(22k) ≤2k∑

i=2

2i

i− 1=

k∑

i=2

2i

i− 1+

2k∑

i=k+1

2i

i− 1

afoÔ π(22) = 2 < 22 = 4.

All�k∑

i=2

2i

i− 1<

k∑i=2

2i kai2k∑

i=k+1

2i

i− 1<

2k∑i=k+1

2i

kkai sunep¸c

π(22k) <k∑

i=2

2i +2k∑

i=k+1

2i

k< 2k+1 +

22k+1

k


afoÔk∑

i=02i = 2k+1 − 1 kai

2k∑i=0

2i

k=

22k+1 − 1k

. EpÐshc èqoume k < 2k kai �ra

2k+1 <22k+1

k, gia k ≥ 1. Sunep¸c

π(22k) < 2

(22k+1)

k

)= 4

(22k

k

)

π(22k)/22k <4k.

T¸ra gia k�je x ≥ 2 up�rqei ènac monadikìc fusikìc arijmìc k tètoioc ¸ste22k−2 < x ≤ 22k. Sunep¸c

π(x)x

≤ π(22k)x

<π(22k)22k−2

= 4(

π(22k)22k

)<

16k

.

Ap� autì prokÔptei ìti gia opoiond pote dosmèno ε > 0 up�rqei o δ = 22([ 16ε ]+1)

tètoioc ¸ste an x ≥ δ tìte k ≥[16

ε

]+ 1 opìte

π(x)x

<16k

<16[

16ε

]+ 1

< ε

afoÔ[16

ε

]+ 1 >

16ε.

'Eqontac autì to apotèlesma, mac dhmiourgeÐtai epiplèon to er¸thma: pwc

fjÐnei to kl�smaπ(x)

x, isodÔnama pwc aux�nei to kl�sma

x

π(x), se sÔgkrish

me to x. Lamb�nontac upìyin ton parak�tw pÐnaka; blèpoume ìti to kl�smax

π(x)aux�nei me polÔ argì rujmì se sqèsh me to x. Eidikìtera, parathroÔme

ìti kaj¸c to x metabaÐnei apì mia dÔnamh tou 10 sthn epìmenh dÔnamh tou 10,

to kl�smax

π(x)aux�nei perÐpou kat� 2, 3. Dhlad to

10k

π(10k)isoÔtai perÐpou

me 2, 3 · k. All� gnwrÐzoume ìti lnx = ln 10 · log x, ìpou lnx kai log x eÐnai olog�rijmoc tou x wc proc th b�sh e(= 2, 7182818 . . . ) kai 10 antÐstoiqa. AutìshmaÐnei ìti ln 10k = (2, 30258 . . . ) ·k, afoÔ ln 10 = 2, 30258 . . . kai log 10k = k.Aut h axioshmeÐwth sumfwnÐa metaxÔ twn tim¸n tou ln x kai

x

π(x)upainÐssetai

th diamìrfwsh thc eikasÐac ìti o lìgoc touc teÐnei sto 1 kaj¸c to x teÐnei sto∞.


Aut h eikasÐa eÐqe diatupwjeÐ apì ton Legendre kai ton Gauss. Sugke-krimèna, to 1778 o Legendre sto biblÐo tou Essai sur la Theorie des Nombres,melet¸ntac th sumperifor� twn tim¸n thc π(x), lamb�nontac upìyin tou autèctic timèc gia x ≤ 400.000 pou tan diajèsimec ekeÐnh thn epoq , diatÔpwse thneikasÐa ìti h sun�rthsh

x

ln x− 1, 08366

eÐnai mia kal prosèggish thc π(x). To 1791 o Gauss se hlikÐa mìlic 14 et¸nmelet¸ntac pÐnakec pr¸twn arijm¸n eÐqe diamorf¸sei thn eikasÐa (all� den thdhmosÐeuse) ìti h π(x) mporeÐ na proseggijeÐ (dhlad aux�nei me ton Ðdio rujmì)

tìso apì th sun�rthshx

ln xìso kai apì th sun�rthsh Li(x) =

x∫2

dt

ln t.

'Ustera apì 100 qrìnia to 1896 o Jacques Hadamard kai o de la Vallee

Poussin anex�rthta o ènac apì ton �llo apèdeixan ìti oi eikasÐec autèc pr�gmatiisqÔoun. Akribèstera, autoÐ apèdeixan ìti

limx→∞

π(x)x/ ln x

= limx→∞

π(x)Li(x)

= 1.

Autì to apotèlesma onom�zetai “To Je¸rhma twn Pr¸twn Arijm¸n” (en sun-tomÐa JPA) kai jewreÐtai èna apì ta spoudaiìtera jewr mata twn majhmatik¸n,ìqi mìno wc apotèlesma autì kaj� eautì all� kai giatÐ gia thn apìdeix touqrei�sthke na anaptuqjoÔn nèoi kl�doi twn majhmatik¸n. To pr¸to b ma giathn apìdeixh aut ofeÐletai ston P. Tchebycheff o opoÐoc apèdeixe to 1850 ìtian up�rqei to ìrio

limx→∞

π(x)x/ lnx

autì prèpei na isoÔtai me 1. Telik� h apìdeixh tou JPA sthrÐqjhke stic idèectou Bernhard Riemann (1826�1866) o opoÐoc anèptuxe th jewrÐa twn miga-dik¸n sunart sewn gia th qr sh thc JewrÐac Arijm¸n pou apoteleÐ s meraènan xeqwristì kl�do thc JewrÐac Arijm¸n thn onomazìmenh Analutik Jew-rÐa Arijm¸n. EpÐshc anafèroume ìti gia polÔ kairì episteÔeto ìti den mporeÐna up�rxei apìdeixh autoÔ tou jewr matoc pou na mh qrhsimopoioÔntai ènnoiecapì th migadik an�lush. EntoÔtoic to 1949 o Paul Erdos kai o Atle Selkerg


anex�rthta o ènac apì ton �llon apèdeixan to JPA qrhsimopoi¸ntac mìno stoi-qei¸deic idiìthtec twn arijm¸n. H apìdeixh aut eÐnai arket� dÔskolh kai demporeÐ na dojeÐ s� autèc tic shmei¸seic.

Shmei¸noume ìti to JPA den shmaÐnei ìti h diafor�∣∣∣π(x) − x

ln x

∣∣∣ (antÐ-stoiqa h |π(x) − Li(x)|) gÐnetai arket� mikr gia meg�la x, all� gÐnetai mikr se sÔgkrish me thn π(x) thn

x

lnx(antÐstoiqa thn Li(x)). 'Eqei de apodei-

qjeÐ ìti h diafor�∣∣∣π(x) − x

lnx

∣∣∣ mporeÐ na p�rei arket� meg�lec timèc. An kaiden eÐnai gnwstì kanèna arijmhtikì par�deigma gia to opoÐo π(x) > Li(x), oJ. Littlewood apèdeixe to 1914 ìti h sun�rthsh π(x)−Li(x) all�zei shmeÐo gia�peirec timèc tou x.

To JPA epÐshc shmaÐnei ìti gia polÔ meg�la x to kl�smaπ(x)

xeÐnai perÐ-

pou1

ln x, dhlad to posostì pou analogeÐ stouc pr¸touc arijmoÔc apì ìlouc

touc arijmoÔc apì to 1 èwc to x eÐnai perÐpou1

lnx. Autì mporeÐ na ekfrasjeÐ

kai wc ex c: An tuqaÐa dialèxoume ènan arijmì metaxÔ tou 1 kai tou x, tìte hpijanìthta na eÐnai autìc pr¸toc isoÔtai me

1ln x

. An antistrèyoume ta kl�-smata, tìte to JPA anafèrei ìti h kat� mèson ìron apìstash (pou orÐzetaiwc to kl�sma

x

π(x)) metaxÔ dÔo diadoqik¸n arijm¸n pou brÐskontai kont� se

ènan arijmì x isoÔtai perÐpou me ln x. Gia par�deigma, ìtan to x eÐnai perÐpou100, epeid ln 100 isoÔtai perÐpou me 4,6, an� pènte arijmoÔc perÐpou, s� autì todi�sthma ja sunantoÔme èna pr¸to. Dhlad , an fantasjoÔme ìti badÐzoume me-taxÔ twn fusik¸n arijm¸n k�nontac èna b ma an� arijmì, tìte euriskìmenoi epÐenìc pr¸tou arijmoÔ p, lìgw tou JPA, gia na fj�soume ston epìmeno pr¸to,ja anamènoume na k�noum ln p b mata.

EpÐshc mporeÐ na deiqjeÐ ìti JPA eÐnai isodÔnamo me to ex c je¸rhma: Anpn eÐnai o n-ostìc pr¸toc arijmìc tìte

limn→∞

n lnn

pn= 1.

Autì shmaÐnei ìti an se èna di�sthma up�rqoun n pr¸toi tìte to m koc toudianÔsmatoc eÐnai perÐpou n ln n (blèpe Hardy and Wright).

Anafèroume epÐshc ìti an�logec eikasÐec èqoun diatupwjeÐ apì ton Hardy


kai Littlewood gia touc dÐdumouc pr¸touc kai gia touc pr¸touc thc morf cx2 + 1. Sugkekrimèna oi eikasÐec autèc anafèroun:

limx→∞

Twin(x)x/(lnx)2

= c kai limx→∞

P (x)√x/ lnx

= c′

ìpou

Twin(x) = |{p ≤ x|p kai p + 2 pr¸toi}|P (x) = |{p ≤ x|p pr¸toc thc morf c n2 + 1}|.

kai oi stajerèc c kai c′ eÐnai perÐpou Ðsec me 0,6601618158 kai 1,3728134628antÐstoiqa.

Pujagìreiec Tri�dec kai h Mèjodoc Kajìdou tou Fermat

'Ena apì ta plèon gnwst� apotelèsmata twn majhmatik¸n eÐnai anamfisb thtato Pujagìreio je¸rhma to opoÐo anafèrei: An x, y, z eÐnai jetikoÐ pragmatikoÐarijmoÐ tìte o z eÐnai to m koc thc upoteÐnousac enìc orjogwnÐou trig¸nou mem kh pleur¸n x, y kai z an kai mìnon an

x2 + y2 = z2.

Fusik� autì to apotèlesma an kei ston kl�do thc GewmetrÐac. An ìmwc jew-r soume th Diofantik exÐswsh x2 + y2 = z2 dhlad anazht soume tic akèraieclÔseic aut c thc exÐswshc, tìte èqoume èna prìblhma pou h lÔsh tou apaiteÐmejìdouc kai ergaleÐa thc JewrÐac Arijm¸n.

Mia tri�da (x, y, z) jetik¸n akèraiwn arijm¸n x, y, z pou ikanopoioÔn thnexÐswsh x2 + y2 = z2 onom�zetai Pujagìreia tri�da (en suntomÐa PT). Giapar�deigma, oi tri�dec (3, 4, 5) kai (5, 12, 13) eÐnai PT. An (x, y, z) eÐnai mia PTtìte gia k�je k ∈ N, h (kx, ky, kz) eÐnai mia PT. Sunep¸c an δ =m.k.d(x, y, z)

kai h (x, y, z) eÐnai PT tìte kai h(x

δ,y

δ,z

δ

)eÐnai PT. Opìte gia ton kajorismì

thc PT arkeÐ na kajorisjoÔn oi PT (x, y, z) me m.k.d(x, y, z) = 1. Autèc ono-m�zontai Prwtarqikèc Pujagìreiec Tri�dec (en suntomÐa PPT). To prìblhma


thc eÔreshc aut¸n eÐqe apasqol sei ton Pujagìra o opoÐoc br ke tic PPT thcmorf c

(2n + 1, 2n2 + n, 2n2 + 2n + 1), n ∈ N.

S mera eÐnai gnwstì ìti polÔ prin ton Pujagìra me to prìblhma autì eÐqanasqolhjeÐ oi Babul¸nioi. Sth dekaetÐa tou 1930 anakalÔfjhke apì ton Neuge-

bauer kai ton Thureau–Dangin mia babul¸nia pinakÐda ìpou èqoun katagrafeÐdÐqwc epexhg seic pollèc PPT. (H pinakÐda aut brÐsketai sth sullog touG.A. Plimpton sto Panepist mio Columbia kai fèrei thn onomasÐa Plimpton

322). Oi PPT eqrhsimopoioÔnto apì touc Babul¸niouc, ìpwc kai mèqri s mera,gia na shmadeÔoun sto èdafoc orjèc gwnÐec pou apantoÔntai sthn anègersh kaikataskeu kthsm�twn.

H genik lÔsh tou probl matoc twn PPT dìjhke gia pr¸th for� sto 10o bi-blÐo twn StoiqeÐwn tou EukleÐdh, o de Diìfantoc sta “Arijmhtik�” anafèretai�mesa èmesa s� aut th lÔsh se di�fora erwt mata pou jètei.

Ja doÔme t¸ra ìti o qarakthrismìc twn PPT mporeÐ na kajorisjeÐ epeid akrib¸c isqÔei to je¸rhma thc monadik c paragontopoÐhshc (dhlad to jeme-lei¸dec je¸rhma thc Arijmhtik c). Sun jwc o qarakthrismìc autìc apodÐdetaimèsw thc tautìthtac tou NeÔtwna:

(m2 − n2)2 + (2mn)2 = (m2 + n2)2, m > n > 0.

'Estw (x, y, z) mia PPT. ParathroÔme ìti

1on m.k.d(x, y) =m.k.d(x, z) =m.k.d(y, z) = 1.

Pr�gmati, an δ|x kai δ|y tìte δ2|x2 kai δ2|y2 opìte δ2|x2 + y2 = z2 kai�ra δ|z (giatÐ;). All� m.k.d(x, y, z) = 1 kai sunep¸c prèpei δ = 1, dhlad m.k.d(x, y) = 1. Me ton Ðdio trìpo prokÔptei ìti m.k.d(x, y) =m.k.d(y, z) =

1.

2on Den mporoÔn oi arijmoÐ x, y na eÐnai kai oi dÔo �rtioi kai oi dÔo perittoÐ,dhlad prèpei 2 - x− y.


Pr�gmati, an oi x, y tan kai oi dÔo �rtioi, tìte den ja tan pr¸toi metaxÔtouc. 'Estw ìti tan kai oi dÔo perittoÐ. Tìte ta tetr�gwn� touc, x2

kai y2, diairoÔmena dia tou 4 ja �fhnan upìloipo 1. Opìte to �jroismax2 + y2 = z2 diairoÔmeno dia 4 ja �fhne upìloipo 2. All� ènac fusikìcarijmìc diairoÔmenoc dia 4 af nei upìloipo 0 1. 'Ara ènac apì toucdÔo x kai y prèpei na eÐnai �rtioc kai o �lloc perittìc. MporoÔme naupojèsoume ìti o x eÐnai �rtioc (diìti diaforetik� all�zoume touc rìlouctwn x kai y), opìte o z− y kai o z + y eÐnai kai autoÐ �rtioi, afoÔ o z kaio y prèpei na eÐnai perittoÐ.

3on T¸ra kaj¸c to x2 diaireÐtai dia 4 mporoÔme na gr�youme

(x

2

)2=

(z + y

2

) (z − y

2

)

kai na efarmìsoume th monadik paragontopoÐhsh shmei¸nontac to ex c.

Oi par�gontecz + y

2kai

z − y

2eÐnai sqetik� pr¸toi metaxÔ touc, afoÔ

k�je koinìc diairèthc touc ja diairoÔse to �jroism� touc pou eÐnai z kaith diafor� touc pou eÐnai y. All� m.k.d(y, z) = 1. Sunep¸c an

z + y

2= pα1

1 · · · pαkk kai

z − y

2= qβ1

1 · · · qβss

eÐnai paragontopoi seic se pr¸touc, to sÔnolo twn pi kai to sÔnolo twn qj

eÐnai xèna metaxÔ touc. Epeid to ginìmeno pα11 · · · pαk

k qβ11 . . . qβs

s isoÔtai me(x

2

)2, dhlad eÐnai tèleio tetr�gwno, prèpei ta αi kai ta βj na eÐnai �rtioi,

me �lla lìgia oi arijmoÐz + y

2kai

z − y

2na eÐnai tèleia tetr�gwna. 'Estw

z + y

2= m2 kai

z − y

2= n2 gia k�poia m,n ∈ N. Afair¸ntac paÐrnoume

y = m2 − n2 kai prosjètontac èqoume z = m2 + n2 kai apì thn x2 =

(z+y)(z−y) prokÔpei x = 2mn. EpÐshc blèpoume ìti m.k.d(m, n) = 1 (kaim > n) afoÔ k�je koinìc diairèthc twn m kai n ja diairoÔse ta x, y kai z.Autì epÐshc mac lèei ìti oi m kai n den mporoÔn na eÐnai kai oi dÔo �rtioi.All� autoÐ den mporoÔn na eÐnai kai oi dÔo perittoÐ, afoÔ diaforetik� kaioi treic arijmoÐ x, y kai z ja tan �rtioi (pou eÐnai adÔnaton).


AntÐstrofa, an m kai n eÐnai jetikoÐ akèraioi, m > n, sqetik� pr¸toimetaxÔ touc kai 2 - m − n, tìte h tri�da (2mn, m2 − n2, m2 + n2) eÐnaimia PPT. Pr�gmati, parathroÔme ìti an δ|m2 − n2 kai δ|m2 + n2 tìteδ|2m2 kai δ|2n2. Epeid m.k.d(m,n) = 1 prèpei δ = 2 1. H perÐptwshδ = 2 apokleÐetai kaj¸c an ènac apì touc m kai n eÐnai �rtioc o �lloceÐnai perittìc.

Sunep¸c èqoume apodeÐxei to ex c.

Je¸rhma. Mia tri�da (x, y, z) me 2|x (antistoiqa 2|y) eÐnai PPT an kai mìnonan up�rqoun jetikoÐ akèraioi m kai n, m > n, me m.k.d(m,n) = 1 kai 2 - m− n

(dhlad o ènac eÐnai �rtioc kai o �lloc eÐnai perittìc), ètsi ¸ste

x = 2mn (antÐstoiqa y = 2mn)

y = m2 − n2 (antÐstoiqa x = m2 − n2)

z = m2 + n2.

Oi fusikoÐ arijmoÐ m kai n onom�zontai genn torec twn PPT.

Parathr seic. 1. An z−x = 1, tìte m2+n2−2mn = (m−n)2 = 1. Opìtem = n + 1 kai �ra x = 2mn = 2n2 + 2n, y = 2n + 1 kai z = 2n2 + 2n + 1. Anenall�goÔn ta x kai y tìte autèc eÐnai oi PPT pou eÐqan kajorisjeÐ apì tonPujagìra.

2. 'Ena endiafèron sumpèrasma thc prohgoÔmenhc melèthc eÐnai to ex c.JewroÔme ènan migadikì arijmì z = m + in, ìpou oi m kai n eÐnai ìpwc stoje¸rhma. 'Eqoume

z2 = m2 − n2 + 2mni kai |z2| = |z|2 = m2 + n2.

Sunep¸c to pragmatikì kai to fantastikì mèroc tou z2 mazÐ me thn apìluth tim tou z2 apoteloÔn mia PPT. Epeid gnwrÐzoume ìti k�je PPT prèpei na èqei thmorf ((m2−n2), 2mn,m2+n2) th morf (2mn,m2−n2,m2+n2), mporoÔmena sumper�noume ìti gia k�je PPT (x, y, z) o migadikìc arijmìc z = x+yi eÐnai


èna tèleio tetr�gwno afoÔ èqei mia tetragwnik rÐza thc morf c m + ni ìpoum kai n eÐnai akèraioi me

m.k.d(m,n) = 1 kai 2 - m− n.

Oi Pujagìreiec tri�dec kai �lla sqetik� probl mata pou anafèrontai sta“Arijmhtik�” tou Diìfantou od ghsan ton Fermat sth diamìrfwsh shmanti-k¸n jewrhm�twn. 'Ena ap� aut� eÐnai gnwstì wc to “TeleutaÐo Je¸rhma touFermat” to opoÐo anafèrei ìti gia k�je fusikì n ≥ 3 den up�rqei kamÐa tri�dajetik¸n akèraiwn x, y, z ètsi ¸ste

xn + yn = zn.

Oi pio gnwstèc shmei¸seic tou Fermat, ap� autèc pou ègrafe sta perij¸ria twnselÐdwn tou antitÔpou “Arijmhtik�” pou meletoÔse, aforoÔn autì to je¸rhma.Sugkekrimèna, o Fermat epishmaÐnei: EÐnai adÔnaton na diaqwrisjeÐ ènac kÔbocse dÔo kÔbouc èna dutetr�gwno se dÔo dutetr�gwna genik� opoiad potedÔnamh, ektìc apì èna tetr�gwno, se dÔo dun�meic me ton Ðdio ekjèth. O Fer-

mat prosjètei ìti “èqw anakalÔyei mia pragmatik� wraÐa apìdeixh autoÔ touapotelèsmatoc all� ìmwc ta perij¸ria twn selÐdwn den èqoun arketì q¸ro giana thn peril�bw”. Gia 350 qrìnia to je¸rhma autì ejewreÐto eikasÐa kai arket�meg�lou kÔrouc majhmatikoÐ afièrwsan arketì qrìno apì th zw touc gia nato apodeÐxoun. Telik� autì apedeÐqjei apì ton Andrew Wiles to 1994 qrhsi-mopoi¸ntac kai sundi�zontac kat�llhla ìlec tic jewrÐec pou eÐqan anaptuqjeÐgia to skopì autì.

An kai h apìdeixh pou isqurizìtan o Fermat ìti eÐqe anakalÔyei den eÐnaignwst (kai �ra den eÐnai sÐgouro ìti tan swst ), up�rqei apìdeixh pou ofeÐ-letai ston Ðdio gia thn perÐptwsh n = 4. Aut sthrÐzetai se mia mèjodo pou oÐdioc epinìhse kai suqn� ef�rmoze prokeimènou na sunag�gei ìti k�poia prìtashpou anafèretai stouc fusikoÔc arijmoÔc den eÐnai alhjin . Aut èqei onomasjeÐ“mèjodoc thc �peirhc kajìdou” kai basÐzetai sthn ex c arq : An h upìjesh ìtimia prìtash anaferomènh stouc fusikoÔc eÐnai alhjin odhg sei me touc kanìnec


thc sunepagwg c sto sqhmatismì miac atèrmonhc kajìdou � dhlad mia fjÐnou-sac diadoq c apì fusikoÔc arijmoÔc sthn opoÐa den up�rqei el�qistoc ìroc �tìte ja èqei prokÔyei esfalmèno sumpèrasma kai, kat� sunèpeia, h upìjesh pouègine den eÐnai paradekt . Gia par�deigma, ac upojèsoume ìti o

√2 eÐnai rhtìc

arijmìc, dhlad √

2 =α

β, α, β ∈ N. Tìte 2β2 = α2, opìte 2|α2 kai �ra 2|α.

Autì shmaÐnei α = 2γ, γ ∈ N kai �ra β2 = 2γ2. Me to Ðdio skeptikì èqoumeβ = 2δ, δ ∈ N, apì to opoÐo èqoume 2δ2 = γ2 dhlad

√2 =

γ

δme γ < α, δ < β

(autèc oi anisìthtec prokÔptoun apì thn α = 2γ kai thn β = 2δ). Autì ìmwcden mporeÐ na isqÔei, giatÐ epanalamb�nontac thn prohgoÔmenh diadikasÐa japaÐrname atèrmonec fjÐnousec diadoqèc fusik¸n arijm¸n α > γ > ε > · · · > 0

kai β > δ > ζ > · · · > 0 ètsi ¸ste√

2 =α

β=

γ

δ=

ε

ζ= · · · . All� oi fusikoÐ

arijmoÐ den mporoÔn na gÐnontai ep� �peiron mikrìteroi kai mikrìteroi ìpwc prin.'Ara h upìjesh ìti o

√2 eÐnai rhtìc den eÐnai paradeqt .

Prèpei na shmei¸soume ìti h “mèjodoc thc kajìdou” eÐnai ousiastik� h“arq tou elaqÐstou” (k�je uposÔnolo twn fusik¸n arijm¸n èqei èna el�qistostoiqeÐo).

ApodeiknÔoume t¸ra ìti h exÐswsh

x4 + y4 = z4

(opìte kai h x4k + t4k = z4k, k ∈ N) den èqei kamÐa akèraia lÔsh x, y, z mexyz 6= 0. Ant� aut c jewroÔme thn exÐswsh

x4 + y4 = x2

apodeiknÔontac ìti aut den èqei kamÐa akèraia lÔsh x, y, z, xyz 6= 0, opìte toÐdio isqÔei kai gia thn prohgoÔmenh

x4 + y4 = (z2)2.

Ac upojèsoume, antijètwc, ìti uparqei mia tètoia lÔsh x, y, z. MporoÔme najewr soume ìti x > 0, y > 0, z > 0 kai epÐshc ìti an� dÔo oi x, y kai z eÐnaisqetik� pr¸toi metaxÔ touc (giatÐ;). Epeid isqÔei (x2)2 + (y2)2 = z2, h tri�da


(x2, y2, z) eÐnai mia PPT. Sunep¸c ènac apì touc x2, y2 prèpei na eÐnai �rtioc,opìte to Ðdio eÐnai kai o x. Tìte up�rqoun m, n ∈ N, m > n, m.k.d(m, n) = 1,o ènac �rtioc kai o �lloc perittìc, tètoioi ¸ste x2 = 2mn, y2 = m2 − n2 kaiz = m2 + n2. An o m eÐnai �rtioc, tìte m2 = 4κ kai n2 = 4λ + 1, κ, λ ∈ N.EÐnai de y2 = 4s + 1, s ∈ N. 'Ara y2 + n2 = 4(s + λ) + 2 = 4κ pou eÐnaiadÔnaton. Sunep¸c o m prèpei na eÐnai perittìc kai o n �rtioc. 'Estw n = 2t,opìte èqoume (

X

2

)2

= mt.

Epeid m.k.d(m, 2t) = 1 kai o m eÐnai perittìc prèpei m.k.d(m, t) = 1. Apì toje¸rhma thc monadik c paragontopoÐhshc prokÔptei ìti m = m2

1 kai t = n21,

m1, n1 ∈ N. T¸ra epeid y2 +n2 = m2 kai m.k.d(y, n, m) = 1, h tri�da (y, n, m)

eÐnai PPT me ton n na eÐnai �rtioc. Opìte up�rqoun m′, n′ ∈ N, m′ > n′,m.k.d(m′, n′) = 1 ètsi ¸ste n = 2m′n′, y = m′2 − n′2 kai m = m′2 + n′2. P�liapì to je¸rhma thc monadik c paragontopoÐhshc, epeid m′n′ =

n

2= n2

1, prèpeim′ = x′2 kai n′ = y′2, x′, y′ ∈ N. Dhlad èqoume

m21 = m = m′2 + n′2 = x′4 + y′4.

'Etsi apì th lÔsh (x, y, z) èqoume brei mia �llh lÔsh (x′, y′,m1) thc x4+y4 = z2

me m1 < z afoÔ 0 < m1 ≤ m21 = m ≤ m2 < m2 + n2 = z. Epanalamb�nontac

thn ìlh diadikasÐa ja sqhmatizìtan mia atèrmonh “k�jodoc” fusik¸n z < m1 <

u < v < · · · qwrÐc el�qisto ìro. 'Ara h upìjesh thc Ôparxhc lÔshc thcx4 + y4 = z2 den eÐnai paradeqt .

Parat rhsh. To prohgoÔmeno je¸rhma mporeÐ na diatupwjeÐ me gewmetri-koÔc ìrouc wc ex c: Den up�rqei orjog¸nio trÐgwno pou ta m kh twn pleur¸ntou eÐnai akèraioi arijmoÐ me aut� twn orj¸n pleur¸n tou na eÐnai tèleia tetr�-gwna.

Shmei¸noume ìti o Fermat katèlhxe sto sumpèrasma ìti h exÐswsh x4+y4 =

z4 den èqei akèraiec lÔseic apodeiknÔontac to ex c prìblhma tou Bachet:“To embadìn enìc orjogwnÐou trig¸nou pou oi pleurèc tou èqoun m kh rh-

toÔc arijmoÔc den mporeÐ na eÐnai Ðso me to tetr�gwno enìc rhtoÔ arijmoÔ”.


Autì to prìblhma an�getai sth melèth thc Diofantik c exÐswshc

x4 − y4 = z2 (∗)

wc ex c. 'Estw ìti up�rqei èna tètoio trÐgwno. Dhlad upojètoume ìti up�r-

qoun rhtoÐ arijmoÐα

α1,

β

β1(pou eÐnai ta m kh twn kajètwn pleur¸n) kai ènac

rhtìcω

ω1ètsi ¸ste

12

( α

α1

β

β1

)=

( ω

ω1

)2.

'Estwγ

γ1to m koc thc upoteÐnousac kai ε = e.k.p(α1, β1, γ1, ω1). Tìte apì

thn exÐswsh( α

α1

)2+

( β

β1

)2=

( γ

γ1

)2paÐrnoume (ε1α)2 + (ε2β)2 = (ε3γ)2 kai

12(ε1α)(ε2β) = (ε4ω)2, ìpou

ε

α1= ε1,

ε

β1= ε2,

ε

γ1= ε3 kai

ε

ω1= ε4. Sunep¸c

mporoÔme na upojèsoume ìti ta m kh twn pleur¸n tou trig¸nou eÐnai jetikoÐakèraioi ìpwc epÐshc ìti to embadìn tou eÐnai to tetr�gwno enìc akèraiou. Dh-lad èqoume tèssereic akeraÐouc α, β, γ, ω pou ikanopoioÔn tic exis¸seic

α2 + β2 = γ2 kai 2αβ = 4ω2.

Ap� autèc paÐrnoume

kai(α + β)2 = α2 + β2 + 2αβ = γ2 + (2ω)2

(α− β)2 = α2 + β2 − 2αβ = γ2 − (2ω)2.

Opìte (α + β)2(α − β)2 = (α2 − β2) = γ4 − (2ω)4. Autì shmaÐnei ìti oi treicarijmoÐ x = γ, y = 2ω kai z = α2 − β2 apoteloÔn mia mh tetrimmènh lÔsh thc(∗). Autì ìmwc den mporeÐ na isqÔei kaj¸c èqoume to ex c.

Je¸rhma. H Diofantik exÐswsh

X4 − Y 4 = Z2 (∗)

den èqei kamÐa (akèraia) lÔsh x, y, z, xyz 6= 0.

Apìdeixh. MporoÔme na upojèsoume ìti x > 0, y > kai z > 0. EpÐshc mpo-roÔme na upojèsoume ìti m.k.d(x, y) = m.k.d(x, z) = m.k.d(y, z) = 1, afoÔ anm.k.d(x, y) = δ, tìte δ2|z2 kai ètsi oi fusikoÐ arijmoÐ

x

δ,

y

δkai

z

δ2eÐnai lÔsh thc


(∗). Me aut thn upìjesh eÔkola prokÔptei ìti o y4 +z2 den diaireÐtai dia 4 kai�ra o x eÐnai perittìc. EpÐshc jewroÔme ìti èqoume mia tètoia lÔsh thc (∗) ìpouo x eÐnai o mikrìteroc jetikìc akèraioc metaxÔ ìlwn aut¸n twn lÔsewn.

T¸ra diakrÐnoume dÔo peript¸seic.

1h PerÐptwsh. O y eÐnai �rtioc. Tìte apì to Je¸rhma twn PPT up�rqounfusikoÐ arijmoÐ m kai n sqetik� pr¸toi metaxÔ touc me 2 - m− n ètsi ¸ste

y2 = 2mn, z = m2 − n2 kai x2 = m2 + n2.

MporoÔme na jewr soume ìti o m eÐnai �rtioc, opìte o n eÐnai perittìc kai�ra m.k.d(2m,n) = 1. Apì to je¸rhma monadik c paragontopoÐhshc prokÔptei,ìpwc kai sta prohgoÔmena jewr mata, ìti 2m = u2 kai n = v2. 'Ara 2|u, èstwu = 2t. Opìte m = 2t2. Sunep¸c èqoume

x2 = m2 + n2 = 4t4 + v4 = (2t2)2 + (v2)2.

P�li apì to je¸rhma twn PPT up�rqoun κ, λ ∈ N, sqetik� pr¸toi me 2 - κ− κ

ètsi ¸ste2t2 = 2κλ, v2 = κ2 − λ2 kai x = κ2 + λ2.

Idiaitèrwc èqoume t2 = κλ, opìte κ = κ21 kai κ = λ2

1. 'Ara v2 = κ41 − λ4

1,dhlad oi fusikoÐ κ1, λ1 kai v apoteloÔn mia �llh lÔsh thc (∗) gia thn opoÐaìmwc èqoume κ1 =

√κ < κ2 + λ2 = x pou eÐnai �topo afoÔ to x èqei jewrhjeÐ

el�qisto.

2h PerÐptwsh. O y na eÐnai perittìc. 'Opwc prin èqoume

z = 2mn, y2 = m2 − n2 kai x2 = m2 + n2.

'Ara m4 − n4 = (m2 − n2)(m2 + n2) = yx)2. Dhlad oi fusikoÐ arijmoÐ m,n

kai yx apoteloÔn lÔsh thc (∗) pou ìmwc isqÔei

m <√

m2 + n2 = x

pou p�li eÐnai �topo.


Ap� autì to je¸rhma prokÔptei �mesa ìti h x4 + y4 = z4 den èqei mh tetrim-mènh lÔsh afoÔ aut an grafteÐ wc z4 − y4 = (x2)2 eÐnai thc morf c (∗).Pìrisma.H Diofantik exÐswsh

x4 + y4 = 2z2

den èqei kamÐa jetik akèraia lÔsh x, y, z, xyz 6= 0, ektìc apì thn x = y = z =

1.

Apìdeixh. 'Eqoume (x4 + y4)2 = x8 + y8 + 2x4y4 = 4z4. Opìte

z4 − (xy)4 =(

x4 − y4

2

)2

.

Parat rhsh. h basik idèa thc teqnik c pou akolouj same gia th lÔsh twnprohgoÔmenwn Diofantik¸n exis¸sewn tan h metatrop thc prosjetik c mor-f c touc se mia pollaplasiastik c morf c aut c oÔtwc ¸ste na eÐnai dunatìnna efarmosjeÐ kat�llhla to je¸rhma thc monadik c paragontopoÐhshc. Giapar�deigma h exÐswshc x2 + y2 = z2 metatrapeÐ sthn pollaplasiastik morf (x

2

)2=

x + y

2· z − y

2. 'Otan aut h teqnik den mporeÐ na efarmosjeÐ tìte ènac

dunatìc trìpoc antimet¸pishc thc melèthc twn exis¸sewn eÐnai na epekteÐnoumetouc akèraiouc arijmoÔc se sust mata arijm¸n sta opoÐa na isqÔei an�logoje¸rhma monadik c paragontopoÐhshc. H melèth tètoiwn susthm�twn od ghsesthn an�ptuxh thc Algebrik c JewrÐac Arijm¸n pou eÐnai pèra apì to skopìautoÔ tou biblÐou.

Ask seic

1. Na deiqjeÐ ìti 3 - n2 + 1 gia k�je n ∈ N.

2. Poioi apì touc arijmoÔc

n4 − 4, n2 + 8n + 7, n4 − 1, n2 − 2n

diairoÔntai dia 5 ìtan o 5|n + 2, n ∈ N.


10. Na deiqjeÐ ìti an 3|α2 + β2 tìte 3|α kai 3|β.

11. AfoÔ deiqjeÐ ìti k�je akèraioc eÐnai thc morf c 3k, 3k + 1 3k + 2, tìteefarmìzontac to gegonìc autì na apodeiqjeÐ ìti o

√3 eÐnai �rrhtoc.

12. Na deiqjeÐ ìti 24|n2 + 23 an 2 - n kai 3 - n ìpou n ∈ N.

13. Na deiqjeÐ ìti 1 + 2 + · · ·+ n|3(12 + 22 + · · ·n2).

14. Na deiqjeÐ ìti kanènac akèraioc thc morf c 11, 111, 1111, . . . , 11111, . . .

den eÐnai tèleio tetr�gwno.

15. Na deiqjeÐ ìti kanènac akèraioc thc morf c 4k + 3 eÐnai to �jroisma dÔotetrag¸nwn.

16. ApodeÐxte ìti gia k�je n ∈ N isqÔei

i) 2|n2 − n

ii) 6|n3 − n

iii) 30|n5 − n

iv) 360|n2(n2 − 1)(n2 − 4)

v) ApofanjeÐte an isqÔei 4|n4 − n, 5|n5 − n, 6|n6 − n k.lp. MporeÐtena eik�sete gia poiouc arijmoÔc k isqÔei k|nk − n, n ∈ N;

17. An n eÐnai perittìc na deiqjeÐ ìti

i) 240|n5 − n

ii) 24|n3 − n.

EpÐshc an kai o m eÐnai perittìc tìte deÐxte ìti 8|m2−n2 kai 8|m4+n4−2.

18. DeÐxte ìti gia k�je n ∈ N, up�rqei m ∈ N ètsi ¸ste k�je ìroc thcakoloujÐac m + 1,mm + 1,mmm

+ 1, . . . diaireÐtai dia n.


19. DeÐxte ìti gia k�je n ∈ N isqÔei

2n−1

∣∣∣∣n∑

i=1

(n

i

)

20. 'Estw n ≥ 2 kai k ∈ N. Na deiqjeÐ ìti (n − 1)2|nk − 1 an kai mìnon ann− 1|k.

21. DeÐxte ìti 4 - n2 + 2 gia k�je n ∈ Z.

22. 'Estw α, β, γ ∈ Z me β > 0, γ > 0. Upojètoume ìti apì thn EukleÐdeiadiaÐresh èqoume

α = βπ + v 0 ≤ v < β

π = γπ′ + v′ 0 ≤ v′ < γ.

Na deiqjeÐ ìti to upìloipo thc diaÐreshc tou α dia tou βγ eÐnai βv′ + v

kai to phlÐko eÐnai π′.

23. Na deiqjeÐ ìti up�rqei 1− 1 antistoiqeÐa metaxÔ twn diairet¸n tou n poueÐnai megalÔterh tou

√n kai ìlwn twn parast�sewn tou n wc diafor� dÔo

tetrag¸nwn, ìpou n ∈ N.

24. Na deiqjeÐ ìti gia k�je akèraio k up�rqei ènac akèraioc n tètoioc ¸ste5|n3 + k. IsqÔei to Ðdio an antÐ tou 5 jewr soume to 7;

25. Upojètoume ìti diairoÔmenoi oi arijmoÐ 1059, 1417 kai 2312 dia enìc arij-moÔ δ > 1 af noun upìloipo v. Na brejeÐ h diafor� δ − v.

26. Na deiqjeÐ ìti o 22n+ 1 diaireÐ ton 222n

+1 − 2.

27. Dojèntwn γ, δ ∈ N na deiqjeÐ ìti up�rqoun α, β ∈ N, ètsi ¸ste (α, β) = δ

kai α + β + γ an kai mìnon an δ|γ.

28. DeÐxte ìti isqÔei (α, β) = (α, β, αx + βy) gia k�je α, β, x, y ∈ Z.

29. DeÐxte ìti up�rqoun �peira to pl joc zeÔgh akèraiwn α, β me (α, β) = 5

kai α + β + 45, en¸ den up�rqei kanèna tètoio zeÔgoc me (α, β) = 3 kaiα + β = 46.


30. Na deiqjeÐ ìti

i) (α, β)|(α + β, α− β)

ii) (α + β, α− β) = 1 2 an (α, β) = 1

iii) (α + β, 4) = 4 an (α, 4) = 2 kai (β, 4) = 2

iv) (α + αβ, β) = 1 an (α, β) = 1.

31. Na brejeÐ to E.K.P., ìpou n ∈ N.

i) [n, n + 1)

ii) [9n + 8, 6n + 5]

32. Na brejoÔn ìla ta m.n ∈ N me m ≥ n tètoia ¸ste (m,n) = 10 kai[m,n] = 100.

33. Na brejoÔn ìlec oi tri�dec α, β, γ ∈ N me α ≥ β ≥ γ, tètoiec ¸ste(α, β, γ) = 10 kai [α, β, γ] = 100.

34. 'Estw α 6= β akèraioi. Na deiqjeÐ ìti up�rqoun �peiroi to pl joc akèraioix ètsi ¸ste

(α + x, β + x) = 1.

35. DeÐxte ìti an γ|αβ tìte γ|(γ, α)β.

36. Upojètoume ìti (α, β) = 1. Na deiqjeÐ ìti

i) (α, β, α− β) = 1 2

ii) (2α + β, α + 2β) = 1 3

iii) (α + β, α2 + β2) = 1 2

iv) (α + β, α2 − αβ + β2) = 1 3

37. Na brejoÔn jetikoÐ akèraioi α kai β tètoioi ¸ste α + β = 5432 kai[α, β] = 223020.


38. Na brejoÔn dÔo akèraioi α kai β ètsi ¸ste α2 + β2 = 85113 kai [α, β] =

1764.

39. Na brejeÐ to [23!41!, 29!37!].

40. Na deiqjeÐ ìti an αn kai βn eÐnai akèraioi pou orÐzontai apì th sqèshαn + βn

√2 = (1 +

√2)n tìte (αn, βn) = 1.

41. DeÐxte ìti gia k�je n ∈ N, up�rqei èna pollapl�sio tou n pou h dekadik tou morf perièqei mìno mon�dec kai mhdenik�. Epiplèon deÐxte ìti an o n

eÐnai sqetik� pr¸toc proc to 10, tìte up�rqei èna pollapl�sio tou n pouh dekadik tou morf perilamb�nei mìno mon�dec.

42. 'Estw n ∈ N kai S ⊆ {1, 2, . . . , 2n} me |S| = n + 1. DeÐxte ìti

i) Up�rqoun α, β ∈ S me (α, β) = 1

ii) Up�rqoun α, β ∈ S me α|β.

43. 'Estw α > 2, β > 2 dÔo fusikoÐ arijmoÐ. DeÐxte ìti o 2β − 1 den diaireÐton 2α + 1.

44. Na brejeÐ o m.k.d. twn α kai β kai na ekfrasjeÐ sth grammik morf αx + βy, x, y ∈ Z.

i) α = 6409, β = 42823

ii) α = 2437, β = 51329

iii) α = 1769, β = 2378.

45. DeÐxte ìti o α2n+ 1 diaireÐ ton α2m − 1 an m > n. EpÐshc deÐxte ìti an

α, m, m ∈ N, m 6= n tìte (α2m+1, α2n

+1) =

{1 an o α eÐnai �rtioc2 an o α eÐnai perittìc.

46. An α, β, n > 1, deÐxte ìti o αn − βn de diaireÐ ton αn + βn.

47. Na deiqjeÐ ìti o rhtìc arijmìc21n + 414n + 3

eÐnai an�gwgo kl�sma, gia k�jen ∈ N.


48. Na brejoÔn dÔo kl�smata me paranomastèc 11 kai 13 antÐstoiqa, ètsi¸ste to �jroism� touc na isoÔtai me

67143

.

49. An m eÐnai perittìc arijmìc tìte deÐxte ìti gia k�je n ∈ N isqÔei (2m −1, 2n + 1) = 1.

50. DeÐxte ìti an α, β ∈ N meα2 + β2

1 + αβ∈ N tìte

α2 + β2

1 + αβ= δ2 gia k�poio

δ ∈ N.

51. Na deiqjeÐ ìti gia k�je n ∈ N, o arijmìc 1 + 2 + 3 + · · · + n diaireÐ ton1m + 2m + 3m + · · ·+ nm, gia k�je perittì m.

Kef�laio 2

Arijmhtik UpoloÐpwn

'Eqei gÐnei dh antilhptì ìti h lÔsh poll¸n problhm�twn diairetìthtac basi-zìntan ìqi tìso stouc arijmoÔc autoÔc kaj� eautoÔc all� sta upìloipa pouaf noun diairoÔmenoi di� enìc stajeroÔ arijmoÔ. ParadeÐgmatoc q�rin, sto prì-blhma: “na brejoÔn ìloi oi pr¸toi arijmoÐ p gia touc opoÐouc oi arijmoÐ p + 10

kai p + 14 eÐnai pr¸toi” h lÔsh sthrÐzetai sto gegonìc ìti ta upìloipa pouaf noun autoÐ oi arijmoÐ diairoÔmenoi dia 3 eÐnai (an p 6= 3) 2 kai 0 0 kai 1antÐstoiqa an o p diairoÔmenoc dia 3 af nei upìloipo 1 2. 'Ara h monadik perÐptwsh gia thn opoÐa h tri�da p, p+10, p+14 apoteleÐtai apì pr¸touc eÐnaip = 3.

Mia �llh kathgorÐa problhm�twn pou h lÔsh touc sqetÐzetai me ta upìloipathc diaÐreshc di� enìc arijmoÔ eÐnai aut� pou anafèrontai sthn Ôparxh mh lÔshcmiac Diofantik c exÐswshc. P.q., èstw f(x, y) èna polu¸numo wc proc x kai y

me akèraiouc suntelestèc. JewroÔme to polu¸numo fn(x, y) pou paÐrnoume apìto f(x, y) antikajist¸ntac touc suntelestèc tou me ta upìloipa pou af noun oiantÐstoiqoi suntelestèc tou f(x, y) diairoÔmenoi dia tou fusikoÔ arijmoÔ n. An(x0, y0) eÐnai mia akèraia lÔsh thc Diofantik c exÐswshc f(x, y) = 0, tìte giak�je fusikì arijmì n up�rqei ènac zn ∈ Z ètsi ¸ste h (vn(x0), vn(y0), zn) eÐnailÔsh thc fn(x, y) = nz, ìpou vn(x0), vn(y0) eÐnai ta upìloipa thc diaÐreshc toux0 kai y0 antÐstoiqa dia tou n. Dhlad , an h f(x, y) = 0 èqei mia akèraia lÔsh,

83

84 Kef�laio 2. Arijmhtik UpoloÐpwn

tìte gia k�je n h fn(x, y) = nz èqei lÔsh. Autì to aplì apotèlesma mporeÐ naqrhsimopoihjeÐ gia na apodeiknÔoume ìti orismènec Diofantikèc exis¸seic denèqoun lÔseic. Pr�gmati, autì mporeÐ na anadiatupwjeÐ isodÔnama wc ex c: Angia k�poio fusikì arijmì n h fn(x, y) = nz den èqei lÔsh tìte kai h f(x, y) = 0

den èqei lÔsh. FereipeÐn, h x2 − 7y − 10 = 0 den èqei lÔsh, giatÐ an aut eÐqelÔsh thn (x0, y0), x0, y0 ∈ Z, tìte o n = 7 ja diairoÔse to x2

0 − 7y0 − 10 isodÔnama to v7(x0)2 − 3. All� oi dunatèc timèc tou v7(x0)2 eÐnai 7k, 7k + 1,7k + 2 kai 7k + 4, k = 0, 1, 2, . . . kai sunep¸c to 7 de diaireÐ to v7(x0)2 − 3.'Ara h x2 − 7y − 10 = 0 den mporeÐ na èqei kamÐa akèraia lÔsh.

O Carl Friedrich Gauss (1777–1855) to 1801, ìtan tan mìlic 24 qrìnwn!dhmosÐeuse thn ergasÐa tou Disquisitiones Arithmeticae (Arijmhtikèc Anaka-lÔyeic). EkeÐ anaptÔssei kai jemelei¸nei th jewrÐa thc arijmhtik c twn upo-loÐpwn. Aut h jewrÐa aplousteÔei kat� ousiastikì trìpo touc arijmhtikoÔcupologismoÔc all� kai tic diatup¸seic kai apodeÐxeic problhm�twn thc JewrÐacArijm¸n. JewreÐtai de to ènausma gia th dhmiourgÐa thc sÔgqronhc �lgebrac.

2.1 IsotimÐec

To kÔrio ergaleÐo pou eis gage o Gauss sth jewrÐa thc arijmhtik c twn upo-loÐpwn eÐnai h ènnoia twn isotimi¸n epÐ tou sunìlou twn akeraÐwn arijm¸n Z.

2.1.1 Orismìc. 'Estw m ènac akèraioc arijmìc. DÔo akèraioi α kai β ja lèmeìti eÐnai isìtimoi ( isoupìloipo) modulo m kai ja gr�foume α ≡ β mod m,an o m diaireÐ th diafor� α − β. Sthn perÐptwsh pou m - α − β ja gr�foumeα 6≡ β mod m.

Gia par�deigma, an m = 7, oi akèraioi α = 13 kai β = −36 eÐnai isìtimoimodulo 7 afoÔ 13− (−36) = 49 = 7 · 7, dhlad me to prohgoÔmeno sumbolismìèqoume 13 ≡ −36mod 7.

EpÐshc èqoume 650 ≡ 13mod 7 kai 624 ≡ −13mod 7, en¸ 11 6≡ 13mod 7.

2.1. IsotimÐec 85

Parathr seic.

1. Epeid m|α − β an kai mìnon an −m|α − β, dhlad α ≡ β modm an kaimìnon an α ≡ β mod (−m), mporoÔme na upojèsoume (kai ètsi ja k�noumesto ex c) ìti o m eÐnai mh arnhtikìc.

2. An m = 0, tìte α ≡ β mod 0 shmaÐnei ìti 0|α−β pou shmaÐnei ìti α−β =

0, dhlad α = β. 'Ara dÔo akèraioi eÐnai isìtimoi modulo 0 an kai mìnonan autoÐ eÐnai Ðsoi.

An m = 1, tìte dÔo akèraioi α, β eÐnai isìtimoi mod 1, afoÔ p�nta isqÔei1|α− β.

(Sto ex c ja upojètoume ìti m > 1 afoÔ oi dÔo peript¸seic m = 0 kai m = 1

den parousi�zoun endiafèron).

H pr¸th idiìthta pou aporrèei apì ton Orismì 2.1.1 kai qarakthrÐzei autìnmèsw twn upoloÐpwn thc diairèsewc tou EukleÐdh eÐnai h ex c.

2.1.2 Prìtash.

i) IsqÔei α ≡ β mod m an kai mìnon an o α kai o β af noun to Ðdio upìloipoìtan diairejoÔn dia tou m.

ii) 'Estw α ∈ Z. Tìte up�rqei monadikìc akèraioc v me α ≡ v mod m kai0 ≤ v < m.

Apìdeixh. i) Apì thn EukleÐdeia diaÐresh èqoume

α = mπ1 + v1, 0 ≤ v1 < m

β = mπ2 + v2, 0 ≤ v2 < m.

ìpou oi π1, π2, v1 kai v2 eÐnai monadikoÐ akèraioi.Sunep¸c α − β = m(π1 − π2) + v1 − v2 kai �ra m|α − β an kai mìnon an

m|v1 − v2. All� |v1 − v2| < m. 'Ara m|α− β an kai mìnoi an v1 = v2.ii) Autì prokÔptei apì th monadikìthta tou upoloÐpou v sthn EukleÐdeia

diaÐresh α = mπ + v, 0 ≤ v < m.


Apì ton Orismì 2.1.1 prokÔptei ìti an α ∈ Z, tìte oi akèraioi pou eÐnaiisìtimoi modulo m me ton α eÐnai akrib¸c ta stoiqeÐa tou sunìlou Kα = {α +

mk/k ∈ Z}. IsqÔei de α ≡ β mod m an kai mìnon an kα = Kβ . Pr�gmati, anKα = Kβ , epeid α ∈ Kα, èqoume α = β+km, gia k�poio k ∈ Z, opìte m|α−β.AntÐstrofa, an m|α − β tìte β = α + km, gia k�poio k ∈ Z, opìte β ∈ Kα

kai an γ = β + mλ ∈ Kβ , tìte γ = α + m(λ + k) ∈ Kα, dhlad Kβ ⊂ Kα.Me ton Ðdio trìpo paÐrnoume kai Kα ⊂ Kβ , opìte Kα = Kβ . Idiaitèrwc, anα = mπ + v, 0 ≤ v < m, tìte Kα = Kv. 'Ara ìla ta dunat� uposÔnola Kα,α ∈ Z, eÐnai ta K0,K1,K2, . . . , Km−1 kai apì thn Prìtash 2.1.2 ii), aut� eÐnaidi�fora an� dÔo. Epiplèon isqÔei Kv ∩Kv′ = ∅, gia k�je v, v′, 0 ≤ v, v′ < m.Pr�gmati, an up rqe t ∈ Kv ∩Kv′ ja eÐqame t = mk + v = mλ + v′, gia k�poiaκ, λ ∈ Z, opìte m|v − v′ kai ja eÐqame Kv = kv′ pou eÐnai �topo.

Sunep¸c to sÔnolo twn akeraÐwn arijm¸n Z diamerÐzetai akrib¸c sta m

uposÔnola Kv, 0 ≤ v ≤ m− 1, dhlad

Z =⋃

0≤v≤m−1

Kv, Kv ∩Kv′ = ∅, 0 ≤ v, v′ ≤ m− 1.

Sthn gl¸ssa thc jewrÐac sunìlwn aut h diamèrish tou Z orÐzei th sqèsh iso-dunamÐac epÐ tou Z: dÔo akèraioi α, β eÐnai isodÔnamoi an kai mìnon an up�rqeiv ∈ Z, 0 ≤ v ≤ m − 1, ètsi ¸ste α, β ∈ Kv, dhlad α ≡ β modm. Sunep¸ch sqèsh α ≡ β mod m eÐnai mia sqèsh isodunamÐac pou oi kl�seic isodunamÐaceÐnai ta uposÔnola Kv, 0 ≤ v ≤ m− 1. (Shmei¸noume ìti eÔkola ja mporoÔsa-me pr¸ta na deÐxoume ìti h sqèsh eÐnai autopaj c, summetrik kai metabatik ,dhlad ìti eÐnai sqèsh isodunamÐac kai met� na prosdiorÐzame tic orizìmenec apìaut kl�seic isodunamÐac).

2.1.3 Orismìc. Th sqèsh α ≡ β mod m th lème sqèsh isotimÐac modulo m

tic de kl�seic isodunamÐac Kv, 0 ≤ v ≤ m − 1, tic lème kl�seic upoloÐpwn ( isotimÐac) modulo m. To sÔnolo {K0,K1,K2, . . . , km−1} aut¸n twn kl�sewnto sumbolÐzoume me Zn ( me Z/mZ).

Sto ex c thn kl�sh upoloÐpwn modulo m sthn opoÐa an kei o akèraioc α

ja th sumbolÐzoume me [α]. Sunep¸c an α ∈ Kv, tìte Kv = [v] = [α]. 'Otan

2.1. IsotimÐec 87

jèloume na dhl¸soume thn ex�rthsh tou sunìlou [α] apì to m, ja gr�foume[α]m α mod m.

Me èna pl rec sÔsthma antipros¸pwn kl�sewn upoloÐpwn modulo m jaennooÔme èna sÔnolo akeraÐwn {α0, α1, . . . , αm−1} ètsi ¸ste wc proc mia di�taxhna èqoume αi = imod m, 0 ≤ i ≤ m−1. Dhlad k�je akèraioc αi an kei se miakai mìno mia kl�sh upoloÐpwn modulo m. Sunep¸c k�je akèraioc eÐnai isìtimocmodulo m me ènan kai mìnon ènan akèraio αi gia k�poio i = 0, 1, . . . , m− 1.

2.1.4 Prìtash. 'Estw {α0, α1, . . . , αm−1} èna pl rec sÔsthma antipros¸-pwn kl�sewn upoloÐpwn kai α, β dÔo akèraioi me (α,m) = 1. Tìte to sÔnolo{ααi +β/i = 0, 1, . . . , m− 1} eÐnai èna pl rec sÔsthma antipros¸pwn kl�sewnupoloÐpwn modulo m.

Apìdeixh. ArkeÐ na deÐxoume ìti an i 6= j, 1 ≤ i, j ≤ m − 1 tìte ααi + β 6=(ααj +β)modm, dhlad m - ααi−ααj = α(αi−αj). All� epeid (α, m) = 1,apì to L mma tou EukleÐdh, prokÔptei ìti m - α(αi − αj) an kai mìnon anm - αi − αj (dhlad αi 6≡ αj mod m).

ParadeÐgmatoc q�rin, to sÔnolo {0, 1, . . . , m− 1} eÐnai èna pl rec sÔsthmaantipros¸pwn kl�sewn upoloÐpwn modulo m. An o m eÐnai perittìc tìte kai to

sÔnolo{

i−m− 12

/i = 0, 1, . . . , m−1

}=

{−m− 1

2,m− 3

2, . . . ,−1, 0, 1, . . . ,

m− 32

,m− 1

2

}eÐnai epÐshc èna pl rec sÔsthma antipros¸pwn kl�sewn upo-

loÐpwn modulo m, en¸ an o m eÐnai �rtioc tìte to sÔnolo{−m

2,−m− 2

2, . . . ,

−1, 0, 1, . . . ,m− 2

2

}eÐnai èna tètoio sÔsthma. To Ðdio isqÔei kai gia to sÔnolo

{0,m + 1, 2(m + 1), . . . , m2 − 1} afoÔ (m,m + 1) = 1. Gia m = 6, ta sÔnola{0, 1, 2, 3, 4, 5}, {α, 5 + α, 10 + α, 15 + α, 20 + α, 25 + α}, α ∈ Z, eÐnai pl rhsust mata antipros¸pwn kl�sewn upoloÐpwnmodulo m.


2.2 Prìsjesh kai Pollaplasiasmìc stic Iso-timÐec

DeÐqnoume t¸ra ìti oi isotimÐec eÐnai sumbatèc se sqèsh me thn prìsjesh kaiton pollaplasiasmì twn akèraiwn arijm¸n. Autìc akrib¸c eÐnai o lìgoc pou hènnoia twn isotimi¸n eÐnai shmantik .

2.2.1 Je¸rhma. 'Estw αi, βi ∈ Z, i = 1, 2, . . . , n. An αi = βi mod m,i = 1, 2, . . . , n tìte

i)n∑

i=1αi ≡

(n∑

i=1βi

)mod m

ii)n∏

i=1αi ≡

(n∏

i=1βi

)modm.

Apìdeixh. 'Eqoume αi = βi + mti, gia k�poio ti ∈ Z, i = 1, 2, . . . , n. 'Aran∑

i=1αi =

n∑i=1

βi + mn∑

i=1ti, opìte

n∑i=1

αi ≡(

n∑i=1

βi

)mod m. EpÐshc

n∏i=1

αi =n∏

i=1βi + mN ìpou N eÐnai ènac akèraioc arijmìc. 'Ara

n∏

i=1

αi ≡(

n∏

i=1

βi

)modm.

2.2.2 Pìrisma. An α ≡ β mod m kai γ ∈ Z, tìte

i) αn ≡ βn mod m

ii) α + γ ≡ (β ± γ)modm

iii) αγ ≡ βγ mod m

iv) α + mk ≡ β mod m.

Apìdeixh. i) Efarmìzoume epagwg sto n. Gia n = 1 dÐdetai ìti isqÔei α ≡β mod m. Upojètontac ìti αn ≡ βn modm tìte apì to 2.2.1 èqoume αn+1 =

ααn ≡ ββn = βn+1 mod m.

2.2. Prìsjesh kai Pollaplasiasmìc stic IsotimÐec 89

ii) IsqÔei γ ≡ γ mod m kai apì to 2.2.1 paÐrnoume α ± γ ≡ (β ± γ)modm

ìpwc kai αγ ≡ βγ mod m pou eÐnai to iii).To iv) prokÔptei apì to gegonìc ìti mk ≡ 0modm.

2.2.3 Pìrisma. An αi ≡ βi mod m, i = 0, 1, 2, . . . , n kai x ≡ x1 mod m tìten∑

i=0

αixi ≡

(n∑

i=0

βixi1

)modm.

Apìdeixh. Epeid αi ≡ βi mod m kai xi ≡ xi1 mod m èqoume αix

i ≡ βixi1 mod m

opìte kain∑

i=0

αixi ≡

(n∑

i=0

βixi1

)modm.

Parat rhsh. H sumbatìthta twn pr�xewn thc prìsjeshc kai tou pollapla-siasmoÔ gia tic isotimÐec mporeÐ na ermhneujeÐ wc h katoqÔrwsh miac algebrik cdom c gia to sÔnolo Zm.

Pr�gmati, mporoÔme na jewr soume to sÔnolo

Zm = {[0], [1], . . . , [m− 1]}

wc ènan “daktÔlio” me pr�xeic thn prìsjesh kai to pollaplasiasmì orÐzontac

[α] + [β] = [α + β] kai [α][β] = [αβ]

afoÔ, apì to 2.2.1, autèc oi pr�xeic eÐnai “kal� orismènec”. Dhlad an [α] = [α′]

kai [β] = [β′] tìte [α + β] = [α′ + β′] kai [αα′] = [ββ′]. Aut eÐnai h je¸rhshpou uiojeteÐtai sth sÔgqronh �lgebra: o daktÔlioc Zm eÐnai o daktÔlioc phlÐkoZ/mZ tou daktulÐou twn akèraiwn Z dia tou ide¸douc mZ ìpou ta stoiqeÐa toueÐnai oi kl�seic [α] = α + mZ.

Sthn klassik jewrÐa arijm¸n den apaiteÐtai mia tètoia je¸rhsh all� ìlecoi pr�xeic gÐnontai sto sÔnolo {0, 1, 2, . . . , m − 1} modulo m ìtan melet�meto sÔnolo Zm. Bèbaia autì gÐnetai kataqrhstik� afoÔ asqoloÔmeja me kl�-seic isodunamÐac akeraÐwn (pou eÐnai uposÔnola akeraÐwn) kai ìqi me akèraiouc.EntoÔtoic, aut h kat�qrhsh sunhjÐzetai sth klassik jewrÐa arijm¸n.


ParadeÐgmata.

1. Gia k�je pr¸to arijmì p > 3 o p2 + 2 eÐnai sÔnjetoc arijmìc. Pr�gmati,èqoume p ≡ ±1 mod 3 opìte p2 ≡ 1 mod 3 kai �ra p2 + 2 ≡ 3 ≡ 0mod 3,dhlad 3|p2 + 2.

2. Gia k�je fusikì arijmì n, o 22n+ 5 eÐnai sÔnjetoc, afoÔ 2 ≡ −1 mod 3

kai �ra 22n ≡ 1mod 3. Opìte 22n+ 5 ≡ 6 ≡ 0mod 3, dhlad 3|22n

+ 5.

3. Oi isotimÐec mac bohjoÔn na k�noume gr gorouc upologismoÔc me dun�meic.Gia par�deigma, gia na broÔme to upìloipo thc diaÐreshc tou

N = (n2 − 1)1001(n2 + 1)1001

dia tou n efarmìzontac to 2.2.1 èqoume

(n2 − 1)1001 = −1modn kai (n2 + 1)1001 ≡ 1modn

�ra N = (n2 − 1)1001(n2 + 1)1001 ≡ −1modn, dhlad to zhtoÔmenoupìloipo eÐnai n− 1.

'Otan jèloume na upologÐsoume to αn mod n, ènac trìpoc eÐnai na upo-logÐsoume to α ≡ r mod n, 0 ≤ r < m, opìte αn ≡ rn modn, met� naupologÐsoume r2 ≡ r′mod n, 0 ≤ r′ < m, opìte αn ≡ r′rn−2 mod n, met�to r′r mod n k.o.k. Aut h mèjodoc eÐnai qronobìroc afoÔ apaiteÐ n − 1

pollaplasiasmoÔc kai se k�je pollaplasiasmì prèpei na upologÐzoume toapotèlesma mod m. 'Enac pio sÔntomoc trìpoc eÐnai na gr�youme ton n

sth diadik morf (blèpe pio k�tw) n = εt2t +εt−12t−1 + · · ·+ε12+ε020,ìpou εi ∈ {0, 1}, opìte

αn =∏

εi=1

α2imod m.

ParadeÐgmatoc q�rin, ac upologÐsoume to 791 mod100. 'Eqoume 91 =


26 + 24 + 23 + 2 + 20, opìte

791 = 726 · 724 · 723 · 72 · 7= (72)2

5(72)2

3(72)2

272 · 7

= ((72)2)24((72)2)2

2((72)2)2 · 72 · 7.

all� 492 = 2401 ≡ 1mod 100, opìte 791 ≡ 72 · 7mod 100 791 ≡43mod 100, dhlad to upìloipo thc diaÐreshc tou 791 dia tou 100 eÐnai43, dhlad o arijmìc 791 sth dekadik morf ja katal gei sto 43.

Ac efarmìsoume to Ðdio sto 250 mod 7. Epeid 50 = 25 + 24 + 2 èqoume250 = 225

22422 = (222

)23(222

)22 · 22 ≡ 223 · 222 · 22 mod7 afoÔ 222

= 16 ≡2mod 7. Opìte 250 ≡ 223 · 222 · 22 ≡ 22 · 2 · 22 ≡ 4mod 7.

4. Me th bo jeia twn isotimi¸n mporoÔme na lÔsoume probl mata ta opoÐa me�llon trìpo eÐnai dÔskolo na lujoÔn. Gia par�deigma, an anazht soumeto upìloipo thc diaÐreshc tou ajroÐsmatoc

S = 15 + 25 + 35 + · · ·+ 995 + 1005

dia tou 4, arkeÐ na parathr soume ìti k�je arijmìc eÐnai isìtimoc me 0 1 2 3 mod 4 kai apì to 1 èwc to 100 up�rqoun 25 arijmoÐ isìtimoi mek�je èna tètoio upìloipo mod4. Epeid de 3 ≡ −1mod 4, èqoume

S ≡ (25 · 05 + 25 · 15 + 25(2)5 + 25(−1)5) mod 4 ≡ 0mod 4.

'Ara to upìloipo eÐnai to 0 dhlad 4|S.

Me thn Ðdia mèjodo deÐqnoume ìti k�je perittìc fusikìc arijmìc n diaireÐto �jroisma

S = 1n + 2n + 3n + · · ·+ (n− 1)n.

ParathroÔme ìti to �jroisma mporeÐ na grafeÐ

S =k=n−1

2∑

k=1

kn +k=n−1

2∑

k=1

(n− k)n =n− 1

2∑

k=1

(kn + (n− k)n).


All� n− k ≡ −k mod n kai �ra (n− k)n ≡ (−k)n ≡ −kn mod n afoÔ on eÐnai perittìc. 'Ara kn + (n− k)n ≡ 0modn kai sunep¸c S = 0 modn.

5. K�je arijmìc thc morf c 11, 111, 1111, . . . den eÐnai tèleio tetr�gwno.ParathroÔme ìti

02 ≡ 0mod 4

12 ≡ 1mod 4

22 ≡ 0mod 4

32 ≡ 1mod 4.

Sunep¸c k�je tèleio tetr�gwno prèpei na eÐnai isìtimo me 0 1 mod4.Epeid 100 ≡ 0mod 4, k�je tètoioc arijmìc eÐnai isìtimoc me 11 ≡ 3 mod 4

kai �ra den eÐnai tèleio tetr�gwno.

Efarmìzontac to Ðdio skeptikì deÐqnoume ìti k�je arijmìc thc morf c4k + 3 den mporeÐ na grafeÐ wc �jroisma dÔo tèleiwn tetrag¸nwn. Pr�g-mati, sÔmfwna me to prohgoÔmeno par�deigma, to �jroisma dÔo tèleiwntetrag¸nwn eÐnai isìtimo me 0 1 2 mod 4 kai �ra k�je akèraioc 4k +3

(≡ 3mod 4) den mporeÐ na eÐnai to �jroisma dÔo tèleiwn tetrag¸nwn.

6. DeÐqnoume ìti h exÐswsh 3x2 +2 = y2 den èqei akèraiec lÔseic. Pr�gmati,an x0, y0 tan mia lÔsh tìte ja eÐqame y2

0 ≡ 2mod 3. All� èna tèleiotetr�gwno mod 3 eÐnai 0 1 kai �ra autì eÐnai adÔnato. Sunep¸c, ìpwc dh èqoume mnhmoneÔsei, oi isotimÐec mporoÔn na qrhsimopoihjoÔn gia naapodeiknÔoume ìti orismènec Diofantikèc exis¸seic den èqoun lÔsh.

Efarmog . Krit ria Diairetìthtac.

Mia endiafèrousa efarmog twn isotimi¸n eÐnai h eÔresh krithrÐwn b�seitwn opoÐwn mporoÔme na apofanjoÔme an ènac akèraioc arijmìc diaireÐtai apìènan �llon. Aut� ta krit ria exart¸ntai apì to arijmhtikì sÔsthma pou gr�-foume touc arijmoÔc, ìpwc gia par�deigma to dekadikì pou eÐnai s mera to pioeÔqrhsto.


Ac deÐxoume kat� arq�c to ex c: 'Estw β ènac akèraioc > 1. Tìte k�jefusikìc arijmìc n gr�fetai monadik� sth morf

n = αmβm + αm−1βm−1 + · · ·+ α0

ìpou 0 ≤ αi < β, i = 0, 1, . . . ,m.Pr�gmati, diair¸ntac ton m dia tou β èqoume

n = βπ0 + α0, 0 ≤ α0 < β,

ìpou π0 kai α0 eÐnai monadikoÐ.Profan¸c π0 ≥ 0. An π0 ≥ β, xanadiairoÔme

π0 = βπ1 + α1, 0 ≤ α1 < β

opìte

n = β(βπ1 + α1) + α0 = β2π1 + βα1 + α0.

SuneqÐzontac me to Ðdio trìpo paÐrnoume thn austhr� fjÐnousa akoloujÐa n >

π1 > π2 > · · · ≥ 0 pou katal gei sto 0 met� apì èna peperasmèno pl jocdiairèsewn, èstw m, ìpou πm = 0 kai πm−1 = βπm + αm = αm, 0 � αm < β.'Etsi èqoume thn par�stash

n =m∑

i=0

αiβi, 0 ≤ αi < β, i = 0, . . . , m.

Upojètontac t¸ra ìti èqoume dÔo tètoiec ekfr�seic Ðsec me:

n =∑

i=0

γiβi =

`′∑

i=0

γ′iβi, 0 ≤ γi, γ

′i < β,

kai γ` 6= 0, γ′`′ 6= 0. An ` ≥ `′ tìte prosjètoume sto deÔtero �jroisma epiplèonìrouc me mhdenikoÔc suntelestèc kai gr�foume

n =∑

i=0

γiβi =

∑

i=0

γ′iβi,


opìte∑i=0

(γi−γ′i)βi = 0. An tan γj−γ′j 6= 0 gia k�poio j, èstw j o mikrìteroc

tètoioc deÐkthc. Opìte

βj∑

i=j

(γi − γi)βi−j = 0 ∑

i=j

(γi − γi)βi−j = 0

kai telik� èqoume

γj − γ′j = β∑

i=j+1

(γ′i − γi)βi−(j+1).

Sunep¸c β|γj − γ′j . All� |γj − γ′j | < β kai �ra γj = γ′j .

Sumbolismìc. Gr�foume (αnαn−1 · · ·α0)β gia na sumbolÐzoume thn èkfrash

tou n wc to �jroisma n =m∑

i=0αiβ

i to opoÐo onom�zetai par�stash tou n wc

proc th b�sh β.

An β = 10 tìte èqoume th dekadik par�stash pou eÐnai h sun jhc graf twn arijm¸n. An β = 2 3 tìte èqoume thn duadik thn triadik par�stash.Gia par�deigma duadik par�stash tou arijmoÔ 3125(= (3125)10) eÐnai h

(110000110101)2 = 211 + 210 + 25 + 24 + 22 + 20.

Shmei¸noume ìti h prohgoÔmenh apìdeixh eÐnai kataskeuastik gia thn par�sta-sh enìc arijmoÔ wc proc mia b�sh β. To teleutaÐo yhfÐo α0 eÐnai to upìloipothc diaÐreshc tou n dia tou β, to epìmeno yhfÐo α1 eÐnai to upìloipo thc diaÐ-reshc tou π0 dia tou β, k.o.k. FereipeÐn, gia na broÔme th duadik� par�stashtou 3125, to diairoÔme dia 2: 3125 = 2 · 1562 + 1, α0 = 1.


Katìpin, diairoÔme to 1562 dia 2: 1562 = 2 · 781 + 0, α1 = 0

SuneqÐzontac èqoume: 781 = 2 · 390 + 1, α2 = 1

390 = 2 · 195 + 0, α3 = 0

195 = 2 · 97 + 1, α4 = 1

97 = 2 · 48 + 1, α5 = 1

48 = 2 · 24 + 0, α6 = 0

24 = 2 · 12 + 0, α7 = 0

12 = 2 · 6 + 0, α8 = 0

6 = 2 · 3 + 0, α9 = 0

3 = 2 · 1 + 1, α10 = 1

1 = 2 · 0 + 1, α11 = 1

'Enac praktikìc trìpoc eÐnai na jewr soume ta diadoqik� phlÐka xekin¸ntac apìton Ðdio ton arijmì (ParadeÐgmatoc q�rin, sto prohgoÔmeno par�deigma èqoumetouc arijmoÔc 3125, 1562, 781, 390, 195, 97, 48, 24, 12, 6, 3, 1). K�je tètoiocarijmìc eÐnai to

12

tou prohgoÔmenou paraleÐpontac to upìloipo. Me autì tontrìpo paÐrnoume ta yhfÐa se antÐstrofh di�taxh gr�fontac gia k�je arijmì 0 1 an o arijmìc eÐnai �rtioc perittìc antÐstoqa.

Aut h mèjodoc odhgeÐ ston ex c asun jisto pollaplasiasmì dÔo arijm¸nα kai β gnwstì wc Pollaplasiasmì twn R¸sswn Agrot¸n: SqhmatÐzoume dÔost lec arijm¸n. H pr¸th apoteleÐtai apì touc arijmoÔc xekin¸ntac apì ton α

kajènac twn opoÐwn eÐnai to12

tou prohgoÔmenou paraleÐpontac to upìloipo,en¸ h deÔterh st lh apoteleÐtai apì touc arijmoÔc kajènac twn opoÐwn eÐnai odipl�sioc tou prohgoÔmenoÔ tou xekin¸ntac apì ton β. An sth deÔterh st lhdiagr�youme touc arijmoÔc pou antistoiqoÔn se �rtiouc arijmoÔc thc pr¸thcst lhc tìte to �jroisma aut¸n pou mènoun eÐnai to zhtoÔmeno ginìmeno α · β.Gia par�deigma, èstw α = 36 kai β = 11, èqoume


36 11

18 22

9 44

4 88

2 176

1 352

396 = 36 · 11.H apìdeixh sthrÐzetai sto gegonìc ìti

α · β =

(n∑

i=0

αi2i

)β

36 · 11 =(25 + 22

) · 11

An gr�youme dÔo jetikoÔc akèraiouc arijmoÔc parist�nont�c touc wc proc miab�sh β, mporoÔme na touc prosjèsoume kai na touc pollaplasi�soume me tonÐdio trìpo pou ekteloÔme autèc tic pr�xeic ìtan oi arijmoÐ parist¸ntai stodekadikì sÔsthma. Gia par�deigma, èstw α1 = (1354)10, α2 = (278)10 tìteα1 = (10101001010)2 kai α2 = (100010110)2. EpÐshc α1 = (1212011)3 kaiα3 = (101022)3. Opìte èqoume

1354

+ 278

1632

10101001010

+ 100010110

11001100000

1212011

+ 101022

2020110

dhlad α1 + α2 = (1632)10 = (11001100000)2 = (2020110)31354× 278108329478

2708376412

10101001010× 10001011000000000000

1010100101010101001010

0000000000010101001010

0000000000000000000000

00000000000101010010101011011111001011100

1212011× 1010220101022

01010220000000

12120110000000

1212011120121222012


Gia na ekteloÔme autèc tic pr�xeic gr gora kai na apofeÔgoume tuqìn l�jh, eÐnaiqr simo na gnwrÐzoume touc pÐnakec thc prìsjeshc kai tou pollaplasiasmoÔ giatouc arijmoÔc pou eÐnai mikrìteroi thc b�shc β. Gia par�deigma, sto dekadikìsÔsthma autoÐ oi pÐnakec eÐnai h gnwst propaÐdeia pou did�sketai sta pr¸taqrìnia tou sqoleÐou. Gia β = 2 kai 3 èqoume

+ 0 1

0 0 11 1 0

× 0 1

0 0 01 0 1

+ 0 1 2

0 0 1 21 1 2 02 2 0 1

× 0 1 2

0 0 0 01 0 1 22 0 2 1

(Shmei¸noume ìti autoÐ oi pÐnakec orÐzoun th dom twn daktulÐwn Zm).

'Eqontac thn par�stash enìc arijmoÔ wc proc mÐa b�sh β mporoÔme na efar-mìzoume to ex c krit rio diairetìthtac.

2.2.4 Je¸rhma. 'Estw m =n∑

i=0αiβ

i, β,m ∈ N, β > 1.

i) An δ|β kai j < n tìte h dÔnamh δj diaireÐ ton m an kai mìnon an o δj

diaireÐ tonj−1∑i=0

αiβi.

ii) An δ|β − 1, tìte δ|m an kai mìnon an δ

∣∣∣∣n∑

i=0αi.

iii) An δ|β + 1, tìte δ|m an kai mìnon an δ

∣∣∣∣n∑

i=0(−1)iαi.

Apìdeixh. i) Kaj¸c β ≡ 0mod δ tìte βj ≡ 0mod δj kai �ra

n∑

i=0

αiβi ≡

(j−1∑

i=0

αiβi

)mod δj .

ii) 'Eqoume β ≡ 1 mod δ kai �ra βi ≡ 1 mod δ, opìte

n∑

i=0

αiβi ≡

(n∑

i=0

αi

)mod δ.


iii) Epeid β ≡ −1mod δ kai �ra βi ≡ (−1)i mod δ, èqoume

n∑

i=0

αiβi ≡

(n∑

i=0

(−1)iαi

)mod δ.

2.2.5 Pìrisma.

i) 'Enac arijmìc m diaireÐtai dia tou 2 (antÐstoiqa dia tou 5) an kai mìnon anto teleutaÐo dekadikì tou yhfÐo diaireÐtai dia tou 2 (antÐstoiqa dia tou 5).

ii) 'Enac arijmìc m diaireÐtai dia tou 3 dia tou 9 an kai mìnon an to �jroismatwn dekadik¸n yhfÐwn tou diaireÐtai dia tou 3 dia tou 9.

iii) 'Enac arijmìc m = (αnαn−1 · · ·α0)10 diaireÐtai dia tou 11 an kai mìnon an

11∣∣∣∣

n∑i=0

(−1)iαi.

2.2.6 Prìtash. To 7 to 11 to 13 diaireÐ ènan arijmì (αnαn−1 · · ·α0)10

an kai mìnon an to 7 to 11 to 13 diaireÐ to �jroisma

(α2α1α0)10 − (α5α4α3)10 + (α8α7α6)10 − · · · (−1)n(αnαn−1αn−2)10

ìpou αn kai αn−1 mporoÔn na jewrhjoÔn mhdèn.

Apìdeixh. To apotèlesma prokÔptei apì to gegonìc ìti 103 ≡ −1mod 1001 kai1001 = 7 · 11 · 13.

ParadeÐgmata.

1. 'Estw m = (αnαn−1 · · ·α0)10 kai m′ = (α0α1 · · ·αn)10 tìte m − m′ ≡0mod 9.Pr�gmati, m−m′ ≡

(n∑

i=0αi

)−

(n∑

i=0αi

)mod 9 = 0 mod 9.

2. An ènac arijmìc èqei n + 1 yhfÐa me n perittì > 3 kai ìla ta yhfÐa tou

eÐnai Ðsa proc 1, dhlad o rn+1 =10n+1 − 1

9, tìte o arijmìc autìc den

eÐnai pr¸toc. Pr�gmati, o arijmìc autìc ja diaireÐtai dia tou 11 afoÔ to

�jroisman∑

i=0(−1)i = 0. Shmei¸noume ìti metaxÔ twn pr¸twn arijm¸n oi

2.3. Nìmoc Apaloif c kai Antistrèyimec Kl�seic 99

pr¸toi thc morf c rn+1 emfanÐzontai me polÔ mikr suqnìthta. Oi mìnoignwstoÐ pr¸toi arijmoÐ rn+1 gia n < 104 eÐnai oi

√2,√

19,√

23,√

317

kai√

1031. Mia anagkaÐa sunj kh gia na eÐnai o rn+1 pr¸toc eÐnai ìti on + 1 na eÐnai pr¸toc.

3. Na brejeÐ to yhfÐo pou leÐpei apì ton arijmì

52817 · 3212146 = 169655× 15282.

Epeid 5− 2 + 8− 1 + 7 = 17 ≡ 6mod 11 kai

3− 2 + 1− 2 + 1− 4 + 6 ≡ −8 ≡ 3mod 11.

Prèpei na èqoume −(−2 + 8 − 2 + 5 − 1 + x − 5 + 5 − 6 + 9 − 6 + 1) =

−6− x ≡ 7mod 11. Opìte x = −2mod 11 �ra x = 9.

2.3 Nìmoc Apaloif c kai Antistrèyimec Kl�-seic

Jewr mata Wilson–Fermat–Euler

GnwrÐzoume ìti gia touc akèraiouc arijmoÔc isqÔei o nìmoc thc apaloif c: anα, β ∈ Z, tìte isqÔei “αβ = 0, an kai mìnon an α = 0 β = 0”. EÐnai fusikìna anarwthjoÔme an o antÐstoiqoc nìmoc isqÔei kai gia tic isotimÐec, dhlad an isqÔei: “αβ ≡ 0modm an kai mìnon an α ≡ 0modm β ≡ 0 mod m”. Hap�nthsh eÐnai arnhtik . Pr�gmati, gia ènan sÔnjeto fusikì arijmì m = m1m2,m1 6= 1, m2 6= 1, en¸ èqoume m1 6≡ 0modm kai m2 6≡ 0modm, isqÔei m1m2 ≡0modm. ParathroÔme ìmwc ìti an o m tan pr¸toc arijmìc, tìte o nìmoc thcapaloif c isqÔei gia tic isotimÐec modm, afoÔ apì to L mma tou EukleÐdhèqoume ìti αβ ≡ 0modm an kai mìnon an α ≡ 0modm β ≡ 0 mod m.Sunep¸c, s� aut thn perÐptwsh, an upojèsoume ìti α 6≡ 0modm, dhlad m - α isodÔnama (α,m) = 1 (afoÔ o m eÐnai pr¸toc), tìte αβ ≡ 0modm an kaimìnon an β = 0modm. H upìjesh α 6≡ 0modm, ìtan o m eÐnai pr¸toc, mporeÐna antikatastajeÐ sth genik perÐptwsh, gia opoiond pote fusikì arijmì m, me


thn upìjesh (α, m) = 1 afoÔ tìte gnwrÐzoume apì to L mma tou EukleÐdh ìti“an αβ ≡ 0modm tìte β ≡ 0mod m” (Fusik� isqÔei kai to antÐstrofo qwrÐcthn upìjesh (α, m) = 1). Sunep¸c o nìmoc thc apaloif c gia tic isotimÐeceÐnai ousiastik� to L mma tou EukleÐdh pou diatup¸netai sthn orologÐa twnisotimi¸n wc ex c.

2.3.1 Je¸rhma. 'Estw m ènac fusikìc arijmìc kai α, β ∈ Z me (α, m) = 1.An αβ ≡ 0modm tìte β ≡ 0modm.

Parat rhsh. Sthn perÐptwsh α = 0, ja èqoume (α, m) = m = 1 opìteα ≡ β ≡ 0mod 1.

To prohgoÔmeno je¸rhma mporeÐ na diatupwjeÐ genik¸tera wc ex c.

2.3.2 Je¸rhma. 'Estw m ènac fusikìc arijmìc kai α, β ∈ Z. An αβ ≡0modm tìte β ≡ 0mod

m

(α,m).

Apìdeixh. H sqèsh αβ ≡ 0modm, shmaÐnei ìti αβ = mk gia k�poio k ∈ Z,opìte

α

(α,m)β =

m

(α,m)k, dhlad

α

(α, m)β ≡ 0mod

m

(α, m)me

( α

(α,m),

m

(α,m)

)=

1. Sunep¸c, apì to 2.3.1, èqoume β ≡ 0modm

(α, m).

2.3.3 Pìrisma. 'Estw m ènac fusikìc arijmìc kai α, β, γ ∈ Z. An αγ ≡βγ mod m tìte α ≡ β mod

m

(γ, m).

Apìdeixh. Gr�foume thn isotimÐa αγ ≡ βγ mod m wc γ(α − β) ≡ 0modm kaiefarmìzoume to 2.3.2.

Par�deigma. DeÐqnoume ìti o arijmìc tou Fermat F5 = 232+1 eÐnai sÔnjetoc(bl. sel. 33). 'Eqoume 5 · 27 ≡ −1mod 641, �ra 54 · 228 ≡ 1mod 641. All�641 = 54 + 24 dhlad −24 ≡ 54 mod 641. Sunep¸c −24 · 54 · 228 ≡ 54 mod 641

kai epeid (54, 641) = 1, telik� 232 ≡ −1mod 641, dhlad F5 = 232 +1 = 641λ,gia k�poio λ ∈ Z.

MporoÔme t¸ra na ermhneÔsoume th sunj kh (α, m) = 1 se sqèsh me ticisotimÐec modm kai wc ex c. GnwrÐzoume ìti aut h sunj kh eÐnai isodÔnamh me


thn Ôparxh akeraÐwn κ kai λ tètoiwn ¸ste ακ + mλ = 1, dhlad ακ = 1−mλ.Dhlad , sth gl¸ssa twn isotimi¸n mod m, h sunj kh (α,m) = 1 eÐnai isodÔna-mh me thn Ôparxh enìc akèraiou κ tètoiou ¸ste ακ ≡ 1modm. ParathroÔme deìti an κ′ ∈ Z tìte ακ′ ≡ 1modm an kai mìnon an κ ≡ κ′modm. Pr�gmati, anκ ≡ κ′mod m tìte ακ′ ≡ ακ ≡ 1modm. AntÐstrofa, an ακ′ ≡ 1modm tìteafoÔ ακ ≡ ακ′ ≡ 1modm èqoume α(κ− κ′) ≡ 0modm kai epeid (α, m) = 1

prokÔptei ìti κ ≡ κ′mod m. Sunep¸c èqoume

2.3.4 Je¸rhma. 'Estw m ènac fusikìc arijmìc kai α ∈ Z. Tìte isqÔei(α, m) = 1 an kai mìnon an up�rqei mia kl�sh upoloÐpwn β mod m tètoia ¸ste

αβ ≡ 1 mod m.

Epiplèon, sthn perÐptwsh aut h kl�sh β mod m eÐnai monadik kai isqÔeiαβ′ ≡ 1modm gia k�je β′ ∈ Z me β′ ≡ β mod m.

2.3.5 Orismìc. Mia kl�sh α mod m lème ìti eÐnai antistrèyimh an up�r-qei mia kl�sh β mod m tètoia ¸ste αβ ≡ 1modm. Mia antistrèyimh kl�shmod m lègetai kai pr¸th kl�sh upoloÐpwn modm. 'Opwc eÐdame prohgoumè-nwc, h kl�sh α mod m eÐnai antistrèyimh an kai mìnon an (α,m) = 1 kai sthnperÐptwsh aut h kl�sh modm gia thn opoÐa αβ ≡ 1modm eÐnai monadik .Gia to lìgo autì, ìpwc gia touc rhtoÔc arijmoÔc, thn kl�sh β modm th lème“h antÐstrofh” thc α modm kai th sumbolÐzoume me α−1 modm

1α

mod m.

Me thn orologÐa aut o nìmoc thc apaloif c 2.3.1 anadiatup¸netai wc ex c.

2.3.6 Je¸rhma. 'Estw α mod m mia antistrèyimh kl�sh. An αβ ≡ 0modm

tìte β ≡ 0modm.

Par�deigma. 'Estw m = 12. Tìte oi mìnec kl�seic mod12 pou eÐnai an-tistrèyimec eÐnai oi 1mod 12, 5mod 12, 7mod 12 kai 11mod 12. Oi antÐstoi-qec antÐstrofec aut¸n eÐnai 1−1 mod 12 = 1 mod 12, 5−1 mod 12 = 5 mod 12,7−1 mod 12 = 7 mod 12 kai 11−1 mod 12 = 11 mod 12. EpÐshc parathroÔme ìtigia tic �llec upìloipec kl�seic èqoume 3 · 4 ≡ 0mod 12, 2 · 6 ≡ 0 mod 12,


3 · 8 ≡ 0mod 12, 4 · 9 ≡ 0mod 12 kai 10 · 6 ≡ 0mod 12, en¸ to ginìmeno miacopoiad pote 6≡ 0mod m kl�shc mod12 epÐ miac antistrèyimhc kl�shc mod 12

eÐnai 6≡ 0mod 12, ìpwc upodeiknÔetai kai apì to 2.3.6.

Parathr seic.

1. H kl�sh 1modm eÐnai p�nta antistrèyimh afoÔ o eautìc thc eÐnai h an-tistrof thc.

2. An α mod m eÐnai antistrèyimh tìte kai h α−1 mod m eÐnai antistrèyi-mh afoÔ h α mod m eÐnai h antÐstrofh thc, dhlad (α−1)−1 modm =

α mod m.

3. 'Enac trìpoc gia na broÔme ènan antiprìswpo thc antÐstrofhc kl�shcmiac antistrèyimhc kl�shc α mod m eÐnai na efarmìsoume ton EukleÐdeioalgìrijmo prosdiorÐzontac akèraiouc κ, λ ètsi ¸ste ακ + mλ = 1. Giapar�deigma, èstw m = 33 kai α = 10. Epeid (33, 10) = 1 h kl�sh10mod 33 eÐnai antistrèyimh. Apì ton EukleÐdeio algìrijmo èqoume

33 = 3 · 10 + 3

10 = 3 · 3 + 1.

'Ara 1 = 10− 3 · 3 = 10− 3(33− 3 · 10) = 10 · 10− 3 · 33.

'Ara κ = 10, λ = −3 kai sunep¸c h kl�sh 10mod 33 eÐnai h antÐstrofhthc 10 mod 33.

4. An o m = p eÐnai pr¸toc arijmìc tìte kaj¸c (p, α) = 1 gia k�je α,1 ≤ α ≤ p− 1, ìlec oi kl�seic 6≡ 0 mod p eÐnai antistrèyimec.

'Enac �lloc qarakthrismìc twn antistrèyimwn kl�sewn mporeÐ na diatupwjeÐwc ex c.

2.3.7 Prìtash. Mia kl�sh α mod m eÐnai antistrèyimh an kai mìnon anαβ 6≡ 0modm, gia k�je β 6≡ 0modm.


Apìdeixh. 'Estw ìti h α mod m eÐnai antistrèyimh. An αβ ≡ 0modm, giak�poio β ∈ Z, (dhlad m|αβ) tìte, epeid (α, m) = 1, prèpei m|β, dhlad β ≡ 0modm. Sunep¸c den mporeÐ na isqÔei β 6≡ 0 mod m kai αβ ≡ 0 mod m.AntÐstrofa, èstw ìti isqÔei αβ 6≡ 0modm, gia k�je β 6≡ 0modm. 'Estwβ1 mod m,β2 mod m, . . . , βm−1 mod m ìlec oi 6≡ 0modm kl�seic mod m. Epei-d βi−βj 6≡ 0modm, gia k�je i 6= j, i, j = 1, . . . , m−1, apì thn upìjesh prèpeiα(βi − βj) 6≡ 0modm, dhlad αβi 6≡ αβj modm. Autì shmaÐnei ìti oi kl�seicαβi mod m, i = 1, . . . , m−1 eÐnai akrib¸c oi kl�seic βi mod m, i = 1, . . . , m−1,(pijan¸c me �llh di�taxh). 'Ara up�rqei èna i ∈ {1, . . . , m − 1} tètoio ¸steαβi ≡ 1modm, afoÔ mia apì apì tic kl�seic βi mod m eÐnai h 1modm. AutìshmaÐnei ìti h α modm eÐnai antistrèyimh.

Mia �llh apìdeixh eÐnai h ex c: 'Estw ìti h α mod m den eÐnai antistrèyimh,

dhlad (α,m) = δ 6= 1. IsqÔei m

∣∣∣∣α

δm = α

m

δ, opìte α

m

δ≡ 0modm me

m

δ6= 0 modm, �topo.

'Estw t¸ra vi mod m, i = 1, 2, . . . , κ ìlec oi antistrèyimec kl�seic mod m,ìpou 0 ≤ vi ≤ m − 1. Dhlad jewroÔme ìla ta vi me 1 ≤ vi ≤ m − 1

kai (vi,m) = 1. An α ∈ Z me (α, m) = 1 tìte α ≡ vi mod m gia k�poioi = 1, . . . , κ. Pr�gmati, h kl�sh α mod m antiproswpeÔetai apì k�poio v, 0 ≤v ≤ m − 1, dhlad α ≡ v mod m. Sunep¸c apì thn 1.1.12 iii) prèpei (α, m) =

(α, v) = 1, dhlad v ≡ vi mod m gia k�poio i = 1, 2, . . . , κ. Autì mac deÐqneiìti an α1 modm, α2 modm, . . . , αs mod m eÐnai an� dÔo diaforèc antistrèyimeckl�seic kai gia k�je akèraio β me (β, m) = 1 ( isodÔnama gia k�je antistrèyimhkl�sh β mod m) up�rqei k�poio i ∈ {1, . . . , s} me β ≡ αi mod m, tìte s = κ.

2.3.8 Orismìc. 'Ena sÔnolo {α1, α2, . . . , ακ} akeraÐwn lègetai pl rec sÔsth-ma antipros¸pwn pr¸twn kl�sewn modulo m pl rec sÔsthma antipros¸pwnantistrèyimwn kl�sewn modulo m an gia k�je akèraio β me (β, m) = 1 up�rqeiènac kai mìno ènac αi, i ∈ {1, . . . , κ} tètoioc ¸ste β ≡ αi modm.

To pl joc κ twn antistrèyimwn kl�sewn mod m to sumbolÐzoume me ϕ(m)

kai h sun�rthsh ϕ : N→ N, m → ϕ(m), onom�zetai sun�rthsh tou Euler.


Gia par�deigma, ϕ(2) = 1, ϕ(3) = 2, ϕ(4) = 2, ϕ(5) = 4, ϕ(6) = 2, ϕ(7) = 6,ϕ(8) = 4, ϕ(9) = 6, ϕ(10) = 4, ϕ(11) = 10 kai ϕ(p) = p − 1, gia k�je pr¸toarijmì p.

'Opwc ja doÔme pio k�tw h sun�rthsh tou Euler eÐnai èna polÔ qr simoergaleÐo gia th JewrÐa Arijm¸n all� kai gia �llouc kl�douc twn Majhmatik¸n.

2.3.9 L mma. 'Estw {α1, α2, . . . , αϕ(m)} èna pl rec sÔsthma antipros¸pwnantistrèyimwn kl�sewn mod m kai α ∈ Z me (α, m) = 1. Tìte to sÔnolo{αα1, . . . , ααϕ(m)} eÐnai epÐshc èna pl rec sÔsthma antipros¸pwn antistrèyi-mwn kl�sewn modm.

Apìdeixh. Kat� arq�c afoÔ (αi,m) = (α, m) = 1, lìgw thc 1.1.12 vi) èqoumekai (ααi,m) = 1. EpÐshc ìlec oi kl�seic ααi mod m eÐnai an� dÔo di�forec,afoÔ diaforetik� ja eÐqame α(αi − αj) ≡ 0modm gia k�poia i, j, i 6= j, 1 ≤i, j ≤ ϕ(m). All� tìte apì to nìmo thc apaloif c ja prèpei αi ≡ αj mod m,afoÔ (α, m) = 1, pou eÐnai �topo sÔmfwna me thn upìjesh.

EÐqame dei (2.1.4) ìti an β eÐnai ènac opoiosd pote akèraioc kai {α1, . . . , αm}eÐnai èna pl rec sÔsthma antipros¸pwn kl�sewn upoloÐpwn mod m tìte kai to{α1+β, . . . , αm+β} eÐnai to Ðdio. Autì den isqÔei genik� gia ta pl rh sust mataantipros¸pwn antistrèyimwn kl�sewn mod m. IsqÔei ìmwc to ex c.

2.3.10 L mma. 'Estw m = pα11 · · · pακ

κ h an�lush tou m se (an� dÔo di�-forouc) pr¸touc arijmoÔc. An {α1, α2, . . . , αϕ(m)} eÐnai èna pl rec sÔsthmaantipros¸pwn antistrèyimwn kl�sewn modm kai β ∈ Z, tìte to sÔsthma{α1 + β, α2 + β, . . . , αϕ(m) + β} eÐnai èna pl rec sÔsthma antipros¸pwn an-tistrèyimwn kl�sewn mod m an kai mìnon an p1p2 · · · pκ|β.

Apìdeixh. DÐdetai wc �skhsh (blèpe 'Askhsh 9).

Wc mia efarmog thc ènnoiac twn antistrèyimwn kl�sewn modulo m prokÔ-ptei to je¸rhma tou Wilson:


2.3.11 Je¸rhma (Wilson). 'Enac fusikìc arijmìc p > 1 eÐnai pr¸toc an kaimìnon an isqÔei

(p− 1)! ≡ −1mod p.

Apìdeixh. Upojètoume ìti (p − 1)! ≡ −1mod p. An o p den tan pr¸toc tìteautìc èqei ènan gn sio diaÐresh δ, 1 < δ ≤ p− 1, opìte o δ eÐnai ènac apì toucpar�gontec tou (p − 1)! kai sunep¸c o δ|(p − 1)!. All� epeid o p diaireÐ ton(p− 1)! + 1, o δ ja diaireÐ to (p− 1)! + 1 kai �ra o δ ja prèpei na diaireÐ ton 1,dhlad δ = 1 pou eÐnai �topo.

'Ara o p den mporeÐ na èqei gn siouc diairètec kai �ra eÐnai pr¸toc.AntÐstrofa, èstw ìti o p 6= 2, 3 eÐnai pr¸toc. JewroÔme to pl rec sÔsthma

{1, 2, . . . , p−1} antipros¸pwn antistrèyimwn kl�sewn mod p. 'Estw α ènac ap�autoÔc gia ton opoÐo isqÔei α2 ≡ 1mod p, dhlad o antÐstrofoc tou α mod p

eÐnai o eautìc tou. All� h isotimÐa α2 ≡ 1mod p eÐnai isodÔnamh me thn (α −1)(α + 1) ≡ 0mod p. Autì shmaÐnei ìti α ≡ 1mod p α ≡ −1mod p. All�α ∈ {1, . . . , p− 1} sunep¸c α = 1 α = p− 1. Sunep¸c gia α ∈ {1, . . . , p− 1}tìte α2 ≡ 1mod p an kai mìnon an α = 1 α = p − 1. Autì mac lèei ìti anα ∈ {2, . . . , p − 2} tìte o antÐstrofoc tou α mod p ja antiproswpeÔetai apìèna stoiqeÐo tou {2, . . . , p− 2} pou eÐnai di�foro tou α. Sunep¸c sto ginìmeno2·3 · · · (p−2) (pou èqei �rtio pl joc paragìntwn) gia k�je par�gonta α up�rqeiènac par�gontac β 6= α tètoioc ¸ste αβ ≡ 1mod p. Opìte 2 · 3 · · · (p − 2) ≡1mod p. Epeid 1 ≡ 1mod p kai p− 1 ≡ −1mod p, telik� paÐrnoume (p− 1)! ≡−1mod p. Gia p = 2 kai p = 3 profan¸c isqÔei (p− 1)! ≡ −1mod p.

Parat rhsh. An n eÐnai sÔnjetoc fusikìc arijmìc 6= 4 tìte isqÔei (n−1)! ≡0modn. Pr�gmati, èstw n = pq, ìpou p eÐnai pr¸toc. An p 6= q tìte o p kai oq eÐnai par�gontec tou (n− 1)! kai �ra o n|(n− 1)!. An p = q 6= 2 tìte n = p2

kai (n − 1)! = 1 · 2 · · · p · · · 2p · · · (n − 1). Opìte o n|(n − 1)!. Autì dÐnei mia�llh apìdeixh thc miac kateÔjunshc tou Jewr matoc tou Wilson.

An kai to Je¸rhma tou Wilson eÐnai èna krit rio gia na exet�zoume an ènacfusikìc arijmìc n eÐnai pr¸toc sÔnjetoc, epeid to (n − 1)! gÐnetai anexè-lekta meg�loc arijmìc kaj¸c to n aux�nei, autì to krit rio eÐnai sthn pr�xh


anef�rmosto. EÐnai ìmwc qr simo gia jewrhtikèc efarmogèc. 'Opwc ja doÔmet¸ra, mia apì autèc afor� tic tetragwnikèc rÐzec modulo èna pr¸to arijmì p.Kat� arq�c jewroÔme dÔo paradeÐgmata.

'Estw p = 11, tìte(

p− 12

)! =

(11− 1

2

)! = 5!. 'Eqoume 5! = 120 ≡

−1mod 11, opìte (5!)2 ≡ 1mod 11. An p = 13 èqoume(

p− 12

)! = 6! = 720 ≡

5mod 13, opìte (6!)2 ≡ −1mod 13.EpishmaÐnoume th diafor� twn dÔo peript¸sewn. Sthn pr¸th perÐptwsh

jewr¸ntac ton arijmìp− 1

2, katìpin to paragontikì tou, èpeita to tetr�gwno

autoÔ kai met� to isìtimo autoÔ modulo p, paÐrnoume wc ap�nthsh to +1.Sth deÔterh perÐptwsh k�nontac to Ðdio paÐrnoume wc ap�nthsh to −1. Poia

eÐnai h ex ghsh aut c thc diafor�c wc proc to shmeÐo ±; SÔmfwna me to Je¸-rhma tou Wilson èqoume 1 · 2 · 3 · · · 5 · 6 · 7 · · · 10 = 10! ≡ −1mod 11. Diaqw-rÐzontac autì to ginìmeno sth mèsh wc (1 · 2 · · · 5)(6 · 7 · · · 10) kai lamb�nontacupìyin ìti 10 ≡ −1mod 11, 9 ≡ −2mod 11, 8 ≡ −3mod 11, 7 ≡ −4mod 11,6 ≡ −5mod 11, paÐrnoume 6·7·8·9·10 ≡ (−5)(−4)(−3)(−2)(−1) = −5!mod 11.

ParathroÔme ìti to shmeÐo meÐon pro lje apì to gegonìc ìti to pl joc twnparagìntwn sto ginìmeno 6 · 7 · 8 · 9 · 10 eÐnai perittì. EpÐshc shmei¸noume ìtiautì to perittì pl joc eÐnai ousiastik� to pl joc twn akèraiwn apì to 1 èwcto 5, dhlad

11− 12

. PhgaÐnontac pÐsw sto −1 ≡ 10! = (1·2 · · · 5)(6·7 · · · 10) ≡(1 · 2 · · · 5)(−(1 · 2 · 3 · · · 5)) ≡ −(5!)2 mod11, blèpoume ìti (5!)2 ≡ 1mod 11.

T¸ra k�noume to Ðdio pr�gma gia p = 13. Apì thn isotimÐa 12! ≡ −1mod 13

paÐrnoume (1 · 2 · · · 6)(7 · 8 · · · 12) ≡ −1mod 13. All� 12 ≡ −1mod 13, 11 ≡−2mod 13, 10 ≡ −3mod 13, 9 ≡ −4mod 13, 8 ≡ −5mod 13 kai 7 ≡ −6mod 13,opìte 7 · 8 · · · 12 ≡ (−6)(−5) · · · (−1) = +6! mod 13. Ed¸ paÐrnoume +6! epei-d to pl joc twn shmeÐwn meÐon, dhlad 6 =

13− 12

, eÐnai �rtio. Sunep¸c−1 ≡ 12! ≡ (1 · 2 · · · 6)2 = (6!)2 mod13.

ParathroÔme ètsi ìti to pìte paÐrnoume +1 −1 gia to((p− 1

2

)!)2

exar-

t¸ntai apì to e�n top− 1

2eÐnai perittìc �rtioc antÐstoiqa. An o

p− 12

eÐnai

perittìc,p− 1

2= 2m + 1, tìte p = 4m + 3 kai an

p− 12

= 2n tìte p = 4n + 1.


Sunep¸c to ìlo jèma an�getai sthn perÐptwsh e�n o p eÐnai den eÐnai thcmorf c 4n + 3 4n + 1.

GenikeÔontac thn prohgoÔmenh parat rhsh, diatup¸noume t¸ra kai apodeik-nÔoume to ex c

2.3.12 Je¸rhma. 'An o p eÐnai pr¸toc thc morf c 4n + 1, tìte mporoÔme nabroÔme ènan akèraio x tètoion ¸ste x2 ≡ −1mod p. 'Enac tètoioc akèraioc eÐnai

o x =(

p− 12

)!.

An o p eÐnai pr¸toc thc morf c 4n + 3 tìte eÐnai adÔnaton na brejeÐ ènactètoioc akèraioc. Me �lla lìgia, an p eÐnai ènac perittìc pr¸toc arijmìc, tìteup�rqei ènac akèraioc x tètoioc ¸ste x2 ≡ −1mod p an kai mìnon an o p eÐnaithc morf c 4n + 1.

Apìdeixh. 'Opwc sthn prohgoÔmenh parat rhsh èqoume(

1 · 2 · · · p− 12

)(p + 1

2· · · (p− 1)

)= (p− 1)! ≡ −1mod p.

All�

p− 1 ≡ −1mod p, p− 2 ≡ −2mod p, . . . ,p + 1

2≡ −p− 1

2mod p.

'Ara(

p + 12

)· · · (p− 1) ≡ (−1)(−2) · · ·

(−

(p− 1

2

))= (−1)2n

(p− 1

2

)!mod p,

an p = 4n + 1, dhlad anp− 1

2= 2n. 'Etsi gia x =

(p− 1

2

)!, blèpoume ìti

pr�gmati x2 ≡ −1 mod p. An o p eÐnai 4n + 3 tìte op− 1

2= 2n + 1, eÐnai

dhlad perittìc. Upojètoume ìti up�rqei x ∈ Z tètoioc ¸ste x2 ≡ −1mod p,opìte profan¸c p - x. Apì to Je¸rhma tou Fermat pou ja apodeÐxoume amèswcpio k�tw, èqoume xp−1 ≡ 1mod p. All� xp−1 = (x2)

p−12 = (x2)2n+1 kai �ra

1 ≡ xp−1 = (x2)2n+1 ≡ (−1)2nh+1 = −1 mod p, dhlad 1 ≡ −1mod p poushmaÐnei ìti p|2, dhlad p = 2, pou eÐnai �topo afoÔ to 2 den eÐnai thc morf c4n + 3.


Epeid ènac perittìc pr¸toc eÐnai thc morf c 4n + 1 4n + 3 paÐrnoume totelikì sumpèrasma.

Parathr seic.

1. An o p eÐnai thc morf c 4n + 1, tìte kai o x = −(

p− 12

)!, ikanopoieÐ

thn isotimÐa x2 ≡ −1mod p. 'Opwc ja doÔme parak�tw (Je¸rhma Lan-

grange) den up�rqei �lloc akèraioc mh isìtimoc me ±(

p− 12

)!mod p pou

ikanopoieÐ thn x2 ≡ −1 mod p.

2. An o p eÐnai thc morf c 4n+3, tìte ìpwc sthn apìdeixh tou 2.3.13, èqoume(

p + 12

)· · · (p−1) ≡ (−1)(−2) · · ·

(−

(p− 1

2

))= (−1)

p−12

(p− 1

2

)!mod p.

All�p− 1

2= 2n + 1, opìte

(p + 1

2

)· · · (p− 1) ≡ −

(p− 1

2

)!mod p.

'Ara

−1 ≡ (p− 1)! ≡(

p− 12

)!(−

(p− 1

2

)!)

= −((

p− 12

)!)2

mod p

((p− 1

2

)!)2

≡ 1mod p,

dhlad [(p− 1

2

)! + 1

] [(p− 1

2

)!− 1

]≡ 0mod p.

Sunep¸c an o p eÐnai thc morf c 4n + 3, èqoume mia apì tic isotimÐec(

p− 12

)! ≡ ±1mod p.

To pìte èqoume +1 −1 ja to doÔme sto L mma tou Gauss sta tetra-gwnik� upìloipa.

3. GnwrÐzoume ìti up�rqoun �peiroi pr¸toi arijmoÐ. Sunep¸c apì to Jèwrh-ma tou Wilson prokÔptei ìti up�rqoun �peiroi sÔnjetoi arijmoÐ thc morf c


n!+ 1. EÐnai anoiktì prìblhma an up�rqoun epÐshc �peiroi pr¸toi arijmoÐthc morf c n! + 1. Shmei¸noume ìti oi monèc timèc tou n metaxÔ tou 1 kai100 gia tic opoÐec o n!+1 eÐnai pr¸toc eÐnai oi n = 1, 2, 3, 11, 27, 37, 41, 73

kai 77.

4. O isqurismìc tou EukleÐdh ìti to sÔnolo twn pr¸twn eÐnai �peiro sth-rÐzetai sto gegonìc ìti o arijmìc p1p2 · · · pκ + 1, ìpou p1, . . . , pκ eÐnaipr¸toi, èqei ènan pr¸to diairèth pou eÐnai di�foroc twn p1, . . . , pκ. T¸raìloi oi perittoÐ pr¸toi eÐnai thc morf c 4n + 3 kai 4n + 1, to de ginìmenopr¸twn thc morf c 4n + 3 eÐnai arijmìc thc morf c 4n + 1, to ginìmenopr¸twn thc morf c 4n + 1 eÐnai p�li arijmìc thc morf c 4n + 1. 'Enacarijmìc thc morf c 4n + 3 prèpei na eÐnai ginìmeno pr¸twn thc morf c4n + 1 kai 4n + 3. O isqurismìc tou EukleÐdh mporeÐ na efarmosjeÐ giana deÐxoume ìti to sÔnolo twn pr¸twn thc morf c 4n + 3 eÐnai �peiro.Apl� jewroÔme ton arijmì 4Q − 1 ìpou Q = q1q2 · · · qκ eÐnai to ginìme-no pr¸twn qi thc morf c 4n + 3, opìte autìc prèpei na èqei èna pr¸todiairèth thc morf c 4n + 3 di�foro twn qi. 'Omwc autìc o isqurismìc denmporeÐ na efarmosjeÐ gia to sÔnolo twn pr¸twn thc morf c 4n+1. Gia naapodeÐxoume ìti autì to sÔn olo eÐnai �peiro mporoÔme na efarmìsoume toJe¸rhma 2.3.13 wc ex c. Upojètoume ìti autì to sÔnolo eÐnai peperasmè-no kai èstw ìti ta stoiqeÐa tou eÐnai oi pr¸toi p1, p2, . . . , pκ. JewroÔmeton arijmì m = 4P 2 + 1 ìpou P = p1p2 · · · pκ. O m eÐnai perittìc kaiden èqei kanènan pr¸to diairèth thc morf c 4n+1, diìti diaforetik� ènactètoioc ja èprepe na eÐnai ènac apì touc pi (oi opoÐoi den mporoÔn na todiairoÔn). Sunep¸c ènac pr¸toc diairèthc tou m ja prèpei na eÐnai thcmorf c 4n + 3 (o m den mporeÐ na eÐnai pr¸toc afoÔ eÐnai thc morf c4n + 1).

'Estw loipìn ìti q eÐnai ènac pr¸toc diairèthc tou m, opìte m = 4P 2+1 ≡0mod q, dhlad (2P )2 ≡ −1mod q. All� autì sÔmfwna me to Je¸rhma2.3.13 eÐnai �topo. 'Ara to pl joc twn pr¸twn thc morf c 4n + 1 prèpeina eÐnai �peiro kai ìqi peperasmèno ìpwc upojèsame.


5. To Je¸rhma tou Wilson ja mporoÔse na diatupwjeÐ kai wc ex c: 'Enacfusikìc arijmìc p > 1 eÐnai pr¸toc an kai mìnon an (p− 2)! ≡ 1mod p pio genik� ènac akèraioc p megalÔteroc Ðsoc apì k�poio jetikì akèraioκ eÐnai pr¸toc an kai mìnon an (κ− 1)!(p−κ)! ≡ (−1)κ mod p. Autì eÐnaipìrisma tou Jewr matoc tou Wilson (to opoÐo eÐnai h eidik perÐptwsh κ =

1: ArkeÐ na parathr soume ìti (p−1)! = 1 ·2 · · · (p−κ)(p−κ+1) · · · (p−2)(p−1) ≡ (p−κ)!(−1)κ−1(κ−1)!mod p afoÔ p−1 ≡ −1mod p, p−2 ≡−2 mod p, . . . , p − (κ − 1) ≡ −(κ − 1mod p. Ap� autì prokÔptei ìtian p eÐnai pr¸toc thc morf c 4m + 3 kai κ = 2m + 2 tìte (2m + 2 −1)!(4m + 3 − 2m − 2)! = (2m + 1)!2 ≡ 1mod p. Dhlad o p diaireÐ ton(2m + 1)!2 − 1 = ((2m + 1)! + 1)((2m + 1)! − 1). Opìte o p eÐnai ènacpr¸toc par�gontac tou (2m + 1)! + 1 tou (2m + 1)! − 1. EpÐshc op eÐnai ènac pr¸toc par�gontac tou (p − 1)! + 1 kai tou (p − 2)! − 1.Genik� oi pr¸toi par�gontec arijm¸n thc morf c n!± 1 eÐnai dÔskolo naprosdiorisjoÔn. An κ =

p + 12

tìte paÐrnoume to dh gnwstì apotèlesma:(p− 1

2

)! ≡ ±1mod p.

6. Mia �llh endiafèrousa gewmetrik apìdeixh tou Jewr matoc tou Wilson

èqei dwjeÐ apì ton J. Petersen to 1869. Aut eÐnai h ex c. 'Estw tadiadoqik� shmeÐa 1, 2, . . . , p epÐ thc perifèreiac enìc kÔklou, ìpou p ènacfusikìc arijmìc, ètsi ¸ste h apìstash tou i− 1 shmeÐou apì to i shmeÐoeÐnai Ðdia me aut tou i shmeÐou apì to i+1 shmeÐo. JewroÔme to polÔgwno123 · · · p sundèontac to shmeÐo 1 me to shmeÐo 2, to 2 me to 3 k.o.k., to p meto 1. Autì eÐnai to kanonikì kurtì p-gwno. Mia met�jesh α1, α2, . . . , αp

twn shmeÐwn 1, 2, . . . , p, sundèontac to α1 me to α2 k.o.k. to αp me toα1 dÐnei èna �llo polÔgwno all� ìqi kurtì. En¸ up�rqoun p! metajèseick�je polÔgwno mporeÐ na kajorisjeÐ me 2p trìpouc m� autì ton trìpo.Diìti af� enìc sqhmatÐzoume èna polÔgwno xekin¸ntac apì opoiond poteapì touc p arijmoÔc wc pr¸th koruf kai afetèrou k�je polÔgwno anti-stoiqeÐ se dÔo kateujÔnseic afoÔ k�je koruf an kei se dÔo pleurèc kaimporoÔme na sqhmatÐsoume to polÔgwno me kateÔjunsh th mia pleur�


me kateÔjunsh thn �llh pleur�. Sunep¸c paÐrnoumep!2p

=(p− 1)!

2dia-

foretik� polÔgwna. Gia par�deigma, an p = 3 èqoume tic ex c metajèseic(anadiat�xeic arÐjmhshc koruf¸n):

JJ

JJ

-

À ]

1

2 3123

JJ

JJ

-

À ]

2

1 3213

JJ

JJ

-

À ]

3

2 1321

JJ

JJ

-

À ]

1

3 2132

JJ

JJ

-

À ]

3

1 2321

JJ

JJ

-

À ]

2

3 1231

pou ìlec dÐnoun to Ðdio polÔgwno (isìpleuro trÐgwno). Diìti af� enìcwc pr¸th koruf mporoÔme na jèsoume touc arijmoÔc 1, 2 kai 3 kai giakajènan èqoume dÔo kateujÔnseic. An p = 4 paÐrnoume trÐa polÔgwnaapì tic ex c 24 metajèseic.

1 2 1 4 2 3 2 1 3 2 3 4 4 3 4 1

4 3 2 3 1 4 3 4 4 1 2 1 1 2 3 21234 1432 2341 2143 3214 3412 4321 4123

¾

-

6 ?¾

-

6 ?¾

-

6 ?¾

-

6 ?¾

-

6 ?¾

-

6 ?¾

-

6 ?¾

-

6 ?

1 2 1 4 2 3 2 1 3 4 3 2 4 1 4 3

JJ

JJ

-

-]À JJ

JJ

¾

¾

^

Á JJ

JJ

-

-]À JJ

JJ

¾

¾

^

Á JJ

JJ

-

¾]À JJ

JJ

¾

¾

^

Á JJ

JJ

-

-]À JJ

JJ

¾

¾

^

Á3 4 3 2 4 1 4 3 1 2 1 4 2 3 2 11243 1423 2314 2134 3421 3241 4132 4312

1 3 1 3 2 4 2 4 3 1 3 1 4 2 4 2

QQ

Q´

´´

6 6s¼ Q

QQ´

´´? ?k3 Q

QQ´

´´

6 6s¼ Q

QQ´

´´? ?k3 Q

QQ´

´´

6 6s¼ Q

QQ´

´´? ?k3 Q

QQ´

´´

6 6s¼ Q

QQ´

´´? ?k3

4 2 2 4 1 3 3 1 2 4 4 2 3 1 1 31324 1342 2431 2413 3142 3124 4213 4231

An p = 5, an� 10 metajèseic paÐrnoume to Ðdio pent�gwno, gia par�deigmapaÐrnoume to Ðdio pent�gwno apì tic 10 metajèseic:

T¸ra èna kanonikì polÔgwno (dhlad autì pou oi pleurèc tou kai oigwnÐec tou eÐnai ìlec Ðsec) kajorÐzetai pl rwc �pax kai èqoume orÐseidÔo diadoqikèc korufèc tou. ParadeÐgmatoc q�rin, gia p = 3 kai p = 4

èqoume èna kanonikì polÔgwno, to isìpleuro trÐgwno kai to tetr�gwno


antÐstoiqa. Gia p = 5 èqoume dÔo kanonik�:

en¸ gia p = 6 èqoume èna kanonikì to ex�gwno:

en¸ gia p = 7 èqoume trÐa kanonik� ept�gwna:

Genik� up�rqoun12ϕ(p) kanonik� polÔgwna p pleur¸n, ìpou ϕ eÐnai h

sun�rthsh tou Euler. GiatÐ an i eÐnai mia koruf pou sundèetai me thnkoruf j tìte to polÔgwno eÐnai kanonikì an kai mìnon an j ≡ (i+δ)mod p

ìpou (δ, p) = 1. Pr�gmati, an (δ, p) = 1 tìte oi arijmoÐ 0, δ, 2δ, . . . , (p−1)δ

modulo p paristoÔn tic korufèc enìc kanonikoÔ polug¸nou p pleur¸n.FereipeÐn, gia p = 9 kai δ = 2, oi arijmoÐ tδ mod 9, t = 0, 1, 2, . . . , p − 1

eÐnai antÐstoiqa oi 0, 2, 4, 6, 8, 1, 3, 5 kai 7 mod 9 pou paristoÔn tickorufèc tou polug¸nou:

AntÐstrofa, an to polÔgwno eÐnai kanonikì tìte h diafor� i−j = δ, ìpou i

kai j eÐnai diadoqikèc korufèc, prèpei profan¸c na plhreÐ thn prohgoÔmenhsunj kh. An de (δ, p) = 1 me δ <

p

2tìte to polÔgwno {0, δ, 2δ, . . . , (p −


1)δ} eÐnai to Ðdio me to polÔgwno {0, p − δ, 2(p − δ), . . . , (p − 1)(p − δ)}.'Etsi paÐrnoume

12ϕ(p) kanonik� polÔgwna.

Idiaitèrwc paÐrnoume12(p − 1) kanonik� polÔgwna me p korufèc an kai

mìnon an o p eÐnai pr¸toc.

Ta upìloipa (mh kanonik�) polÔgwna omadopoioÔntai an� p polÔgwna,afoÔ kajèna ap� aut� peristrèfontac to gÔrw apì to kèntro tou kÔkloukat� gwnÐa

2πi

p, i = 0, 1, . . . , p− 1, paÐrnei p diaforetikèc jèseic. 'Ara o

p diaireÐ ton(p− 1)!

2− 1

2(p − 1) =

p− 12

((p− 2)!− 1) kai sunep¸c ton(p− 2)!− 1.

Mia apì tic pio qr simec isotimÐec h opoÐa ofeÐletai ston Euler eÐnai h ex c.

2.3.13 Je¸rhma (Euler). Gia k�je fusikì arijmì m kai k�je akèraio α me(α, m) = 1, dhlad gia k�je antistrèyimh kl�sh α modm, isqÔei h isotimÐa

αϕ(m) ≡ 1modm.

Sunep¸c h antÐstrofh kl�sh thc α mod m eÐnai h αϕ(m)−1 mod m.

Apìdeixh. 'Estw {α1, α2, . . . , αϕ(m)} èna pl rec sÔsthma antipros¸pwn anti-strèyimwn kl�sewn mod m. SÔmfwna me to 2.3.9 to sÔnolo {αα1, . . . , ααϕ(m)}eÐnai èna pl rec sÔsthma antipros¸pwn antistrèyimwn kl�sewn mod m. Su-nep¸c gia k�je i = 1, . . . , ϕ(m), up�rqei k�poio j = 1, . . . , ϕ(m), ètsi ¸steααi ≡ αj modm. Pollaplasi�zontac ìlec autèc tic ϕ(m) to pl joc isotimÐecèqoume

αϕ(m)α1α2 · · ·αϕ(m) ≡ α1α2 · · ·αϕ(m) modm

(αϕ(m) − 1)α1α2 · · ·αϕ(m) ≡ 0modm.

Epeid (αi,m) = 1, i = 1, 2, . . . , ϕ(m), èqoume kai (α1α2 · · ·αϕ(m),m) = 1.'Etsi apì to nìmo apaloif c paÐrnoume telik� αϕ(m) ≡ 1mod m.

To je¸rhma autì eÐnai h genÐkeush tou gnwstoÔ wc “to mikrì je¸rhma” touFermat pou anafèrei to ex c.


2.3.14 Je¸rhma (Fermat). 'Estw p ènac pr¸toc arijmìc. Tìte gia k�jeakèraio arijmì α isqÔei

αp ≡ α mod p.

An p - α tìte

αp−1 ≡ 1mod p.

Apìdeixh. DÐnoume duo apodeÐxeic.1h. An p - α, tìte apì to 2.3.13, èqoume αp−1 ≡ 1 mod p afoÔ ϕ(p) = p− 1.

Pollaplasi�zontac aut thn isotimÐa epÐ α paÐrnoume αp ≡ α mod p. All� kaian p|α, èqoume

αp ≡ α ≡ 0mod p.

Opìte gia k�je α ∈ Z isqÔei αp ≡ α mod p.2h. ApodeiknÔoume thn isotimÐa αp ≡ α mod p gia mh arnhtikoÔc akèraiouc

α, qrhsimopoi¸ntac epagwg . Gia α = 0 kai α = 1 h isotimÐa profan¸c isqÔei.Upojètoume ìti isqÔei αp ≡ α mod p kai apodeiknÔoume ìti isqÔei kai (α+1)p ≡(α + 1)mod p. 'Eqoume (α + 1)p =

p∑i=10

(pi

)αi ≡ (αp + 1)mod p, afoÔ isqÔei

(pi

) ≡ 0mod p, gia i, 1 ≤ i ≤ p − 1. Pr�gmati, epeid p(p − 1)! =(pi

)!(p − i)!,

o p diaireÐ ton(pi

) ton i!(p− i)!. All� o p den mporeÐ na diaireÐ ton i!(p− i)!.

All� o p den mporeÐ na diaireÐ ton i!(p − i)!, giatÐ diaforetik� o p ja èprepena diairoÔse ènan arijmì mikrìtero ap� ton p afoÔ o i!(p − i)! eÐnai ginìmenoarijm¸n mikrìterwn tou p.

Epeid upojèsame ìti αp ≡ α mod p, telik� èqoume ìti (α + 1)p) ≡ (α +

1)mod p.An o α eÐnai arnhtikìc tìte up�rqei k�poioc v, 0 ≤ v ≤ p− 1 tètoioc ¸ste

α ≡ v mod p. Opìte αp ≡ vp mod p kai epeid vp ≡ v mod p èqoume telik�αp ≡ α mod p. 'Enac �lloc trìpoc apìdeixhc autoÔ eÐnai o ex c. An p eÐnaiperittìc, èqoume αp = −(−α)p ≡ −(−α)mod p dhlad αp ≡ α mod p. Anp = 2, tìte α2 = (−α)2 ≡ −α mod 2, all� −α ≡ α mod 2.

An p - α, tìte apì ton nìmo apaloif c, paÐrnoume thn isotimÐa αp−1 ≡1mod p.


Mia apì tic efarmogèc tou Je¸rhmatoc tou Euler eÐnai o sqetik� eÔkolocupologismìc tou upoloÐpou thc diaÐreshc meg�lwn dun�mewn arijm¸n di� enìcfusikoÔ arijmoÔ. Gia par�deigma, gia na broÔme to upìloipo thc diaÐreshc tou35506 dia tou 24, ergazìmaste wc ex c.

UpologÐzoume to ϕ(24) pou eÐnai 8. DiairoÔme to 506 dia tou 8, 506 =

8 · 63 + 2 kai èqoume

35506 = (358)63352 = (35ϕ(24))63352 ≡ 352 mod24.

All� 352 = (5 ·7)2 = 52 ·72 = 25 ·49 kai 49 ≡ 25mod 24, 25 ≡ 1 mod 24, opìte

35506 ≡ 1mod 24.

Mia �llh efarmog tou Jewr matoc tou Euler, pou anafèretai kai sth dia-tÔpws tou, eÐnai ìti mac epitrèpei na kajorÐzoume thn antÐstrofh kl�sh miacantistrèyimhc kl�shc mod m. Gia par�deigma, an m = 27, h kl�sh 11mod 27

eÐnai antistrèyimh kai h antistrof thc eÐnai h 11ϕ(27)−1 = 1117 mod 27. EÐnaide 1117 = (114)4 · 11 =

((121)2

)4 · 11 ≡ (132)4 · 11 mod 27. All� 132 = 169 ≡7mod 27 kai 74 ≡ −2mod 27. 'Ara 1117 ≡ −2 · 11 ≡ 5mod 27, pou pr�gmati11 · 5 ≡ 1mod 27.

EpÐshc to Je¸rhma tou Fermat mporeÐ na qrhsimopoihjeÐ gia na elègqoumean ènac arijmìc n eÐnai sÔnjetoc, kaj¸c autì mac lèei ìti an gia k�poio α ∈Z den isqÔei αn ≡ α mod n tìte o n den eÐnai pr¸toc. Gia par�deigma, ann = 143 kai α = 2 paÐrnoume 2143 = 27·2023 = (27)2023 = (128)20 · 23 ≡(−15)20 · 23 mod 143. All� (−15)20 = ((−15)2)10 = 22510 ≡ 8210 mod 143 kai822 = 6724 ≡ 3mod 143. 'Ara 2143 ≡ 35 · 23 mod143. EpÐshc 35 = 243 ≡100mod 143. Sunep¸c 2143 ≡ 800mod 143. Telik� 2143 ≡ 85mod 143 kaiepeid 85 6≡ 2mod 143, o 143 den eÐnai pr¸toc. Fusik� autì to par�deigma eÐnaitetrimmèno afoÔ 143 = 11 · 13, all� deÐqnei th diadikasÐa pou akoloujoÔme.

Shmei¸noume ed¸ ìti to antÐstrofo tou Jewr matoc tou Fermat den isqÔei,kaj¸c to 1910 o R. Carmichael parat rhse ìti gia k�je akèraio α isqÔei α561 ≡amod561 en¸ 561 = 3 · 11 · 17. Argìtera ja apodeÐxoume to ex c krit rio touA. Korselt (1899): “O n diaireÐ ton αn − a gia ìlouc touc akèraiouc a an kai


mìnon an o n den diaireÐtai apì to tetr�gwno opoioud pote pr¸tou arijmoÔ kaio p − 1 diaireÐ ton n − 1 gia ìlouc touc pr¸touc p pou diairoÔn ton n”. (Ap�autì prokÔptei ìti o 561 eÐnai o mikrìteroc n m� aut thn idiìthta). Oi fusikoÐarijmoÐ n gia touc opoÐouc isqÔei an ≡ amod n gia k�je α ∈ Z, lègontai �peiroiarijmoÐ tou Carmichael. 'Ewc to 1994 den tan gnwstì an up rqan �peiroiarijmoÐ Carmichael. To 1994 oi Alford, Granville kai Pomerance apèdeixan ìtipr�gmati to pl joc aut¸n twn arijm¸n eÐnai �peiro.

DÐnoume t¸ra thn apìdeixh tou Jewr matoc tou Euler ìpwc to apèdeixe oÐdioc o Euler to 1760. 'Estw α ènac opoiosd pote akèraioc kai m > 1 ènacfusikìc arijmìc. JewroÔme tic dun�meic α0 = 1, α1, α2, . . . , αm. H “Arq twnPerister¸nwn” anafèrei ìti an n antikeÐmena katanemhjoÔn se m jèseic kain > m tìte toul�qiston dÔo apì aut� ta antikeÐmena ja katal�boun thn Ðdiajèsh. 'Ara metaxÔ twn m + 1 upoloÐpwn thc diaÐreshc twn dun�mewn αk diatou m ja prèpei na up�rqoun toul�qiston dÔo upìloipa ta opoÐa na eÐnai Ðsa,afoÔ ta dunat� upìloipa eÐnai se pl joc m. Dhlad up�rqoun dÔo dun�meicαi kai αj tou α me i 6= j ètsi ¸ste αi ≡ αj mod m. Upojètoume t¸ra ìti(α, m) = 1. Epeid i 6= j (èna ap� aut� eÐnai mikrìtero tou �llou) mporoÔmena upojèsoume ìti i > j. Kaj¸c (α,m) = 1, eÐnai kai (αk,m) = 1, gia k�jek ∈ N. Opìte up�rqei o antÐstrofoc tou αj mod m. Pollaplasi�zontac thnisotimÐa αi ≡ αj mod m epÐ autìn ton antÐstrofo paÐrnoume

αi−j ≡ 1modm.

Epeid o j ≥ 0 kai o i ≤ m ja èqoume 0 < i − j ≤ m. Sunep¸c up�rqei ènacjetikìc akèraioc ν tètoioc ¸ste

αν ≡ 1 mod m kai αµ 6≡ 1modm

gia k�je µ, 1 ≤ µ < ν. Autìn ton arijmì ν ton lème t�xh tou α mod m.Shmei¸noume ìti prohgoumènwc antÐ na jewr soume tic dun�meic α0, α1, α2

, . . . , αm, sthn perÐptwsh pou o α eÐnai pr¸toc proc ton m, mporoÔme na jew-r soume tic dun�meic α0, α1, α2, . . . , αϕ(m) pou k�je mia orÐzei mia pr¸th kl�shupoloÐpwn mod m. Epeid ìlec autèc oi dun�meic eÐnai se pl joc ϕ(m) + 1,


sÔmfwna me thn Arq twn Perister¸nwn, dÔo apì autèc ja sumpÐptoun, afoÔup�rqoun akrib¸c ϕ(m) pr¸tec kl�seic mod m. 'Ara to i ja eÐnai ≤ ϕ(m) kai�ra to ν ja eÐnai ki autì ≤ ϕ(m).

An to ν > 1, dhlad o α 6≡ 1modm, oi arijmoÐ α0, α, α2, . . . , αν−1 eÐnai an�dÔo mh isìtimoi modm, (diìti diaforetik� h t�xh tou α ja tan mikrìterh touν) kai h akoloujÐa twn dun�mewn 1, α, α2, . . . , αν , αν+1, . . . , eÐnai mia periodik akoloujÐa modm, dhlad isqÔei

αk1 ≡ αk2 mod m

an kai mìnon an k1 ≡ k2 mod ν.'Opwc shmei¸same, èqoume ìti ν ≤ ϕ(m). An ϕ(m) > 1, dhlad m 6= 2

kai ν < ϕ(m), tìte ja up�rqei toul�qiston ènac fusikìc arijmìc β sqetik�pr¸toc proc ton m pou eÐnai mh isìtimoc me touc arijmoÔc 1, α, α2, . . . , αν−1 kai�ra mh-isìtimoc me k�je dÔnamh αn gia ìla ta n ≥ 0.

JewroÔme ta ginìmena

β · 1, β · α, β · α2, . . . , βαν−1

ta opoÐa eÐnai mh isìtima mod m me kajèna apì touc arijmoÔc 1, α, α2, . . . , αν−1.Pr�gmati, an tan βαj ≡ αi modm me 0 ≤ j ≤ ν−1, 0 ≤ i ≤ ν−1 tìte, kaj¸cαν ≡ 1modm, ja eÐqame

βαj ≡ αi+ν mod m

kai �raβ ≡ αi+ν−j mod m

pou eÐnai �topo, afoÔ o β den eÐnai isìtimoc me k�je dÔnamh αn mod m. Epiplèonoi arijmoÐ β1, βα, . . . , βαν−1 eÐnai mh-isìtimoi mod m, diìti an tan βαi ≡ βαj

ja eÐqame kai αi ≡ αj mod m me 0 ≤ i 6= j ≤ ν − 1. 'Etsi paÐrnoume touc 2ν

arijmoÔc1, α, . . . , αν−1, β · 1, β · α, . . . , βαν−1

pou eÐnai an� dÔo mh-isìtimoi mod m kai pr¸toi proc ton m. 'Ara ϕ(m) ≥ 2ν.An ϕ(m) = 2ν, tìte αϕ(m) = α2ν = (αν)2 ≡ 1modm. An ϕ(m) > 2ν tìte


epanalamb�noume thn Ðdia diadikasÐa, dhlad mporoÔme na broÔme ènan arijmì γ,(γ, m) = 1, gia ton opoÐon oi arijmoÐ

1, α, . . . , αν−1, β, βα, βα2, . . . , βαν−1, γ, γα, . . . , γαν−1

eÐnai pr¸toi proc ton m kai epÐshc eÐnai an� dÔo mh isìtimoi mod m. 'Etsi jaèqoume ϕ(m) ≥ 3ν.

An ϕ(m) > 3ν tìte αϕ(m)z(αν)3 ≡ 1modm en¸ an ϕ(m) > 3ν suneqÐ-zoume thn Ðdia diadikasÐa. All� o ϕ(m) eÐnai ènac kajorismènoc arijmìc kaiaut h diadikasÐa prèpei na termatisjeÐ, dhlad ja prèpei na up�rqei k�poiopollapl�sio to ν pou na eÐnai Ðso me ϕ(m) èstw ϕ(m) = kν opìte telik�αϕ(m) = (αν)k ≡ 1modm.

Parat rhsh. Ta sÔnola twn arijm¸n {1, α, α2, . . . , αν−1}, {β, βα, . . . , βαν−1},{γ, γα, . . . , γαν−1}, . . . onom�zontai pleurikèc kl�seic modm gia to sÔnolo{1, α, . . . , αν−1}. An de R = {r1, r2, . . . , rϕ(m)} eÐnai èna pl rec sÔsthma anti-pros¸pwn pr¸twn kl�sewn mod m kai r eÐnai èna stoiqeÐo tou R pou h t�xh

tou eÐnai ν tìte ν|ϕ(m). To de R diamerÐzetai seϕ(m)

νuposÔnola pou se èna

apì ta uposÔnola thc diamèrishc aut c perièqontai ta stoiqeÐa tou R pou eÐnaiisìtima proc ta stoiqeÐa tou sunìlou {1, r, . . . , rν−1}. An ϕ(m) > ν, ta upì-

loipaϕ(m)

ν− 1 uposÔnola thc diamèrishc aut c to kajèna perièqoun stoiqeÐa

pou eÐnai isìtima proc ta antÐstoiqa stoiqeÐa twn upìloipwn pleurik¸n kl�sewnmod m tou {1, r, . . . , rν−1}.

Anafèrame thn prohgoÔmenh apìdeixh diìti aut eÐnai ènac pro�ggeloc thcjewrÐac om�dwn. IdiaÐtera h prohgoÔmenh parat rhsh den eÐnai tÐpota �llo par�to Je¸rhma tou Langrange thc JewrÐac twn Peperasmènwn Om�dwn.

Par�deigma. 'Estw m = 21, tìte ϕ(m) = 12 kai to sÔnolo R = {1, 2, 4, 5, 8,

10, 11, 13, 16, 17, 19, 20} eÐnai èna pl rec sÔsthma pr¸twn kl�sewn upoloÐpwnmod21. 'Estw α = 2, tìte α0 = 1, α = 2, α2 = 4, α3 = 8, α4 = 16,α5 = 32 ≡ 11mod 21 kai α6 ≡ 1mod 21. 'Ara h t�xh tou 2 eÐnai 6. Sunep¸c toR diamerÐzetai se dÔo uposÔnola to èna eÐnai to {1, 2, 4, 8, 16, 11} kai to �llo{5, 10, 13, 17, 19, 20} apoteleÐtai apì stoiqeÐa pou eÐnai isìtima me ta stoiqeÐa

2.4. Grammikèc IsotimÐec 119

thc pleurik c kl�shc {10, 20, 40, 80, 160, 320} ìpou ed¸ p rame β = 10. To Ðdioja tan an β = 13 17 19 20.

Wc èna pìrisma tou Jewr matoc tou Euler èqoume to ex c.

2.3.15 Pìrisma. 'Estw α mod m mia antistrèyimh kl�sh modulo m. Ann ≡ r mod ϕ(m) tìte αn ≡ αr mod m.

Apìdeixh. 'Eqoume n = r + kϕ(m), opìte αn = αr+kϕ(m) = αrαkϕ(m) ≡αr mod m.

Parathr seic.

1. Autì to pìrisma isqÔei kai an antÐ tou ϕ(m) jèsoume thn t�xh touα mod m.

2. IsqÔei kai to antÐstrofo autoÔ tou porÐsmatoc, me thn proôpìjesh ìtiαv 6≡ 1modm gia k�je v, 0 < v < ϕ(m). Pr�gmati, an αn ≡ αr mod m,dhlad αn−r ≡ 1modm èstw n− r = kϕ(m) + v, v ≤ v < ϕ(m), opìteαn−r = αkϕ(m)αv. All� αn−r ≡ 1modm kai αkϕ(m) ≡ 1modm, opìteαv ≡ 1mod m �ra v = 0 kai sunep¸c n ≡ r mod ϕ(m). Ed¸ h t�xh touα eÐnai ϕ(m). Genik� an αn ≡ αr mod m tìte n ≡ r mod ν ìpou ν eÐnai ht�xh tou α modm.

Par�deigma. Na brejeÐ to upìloipo thc diaÐreshc tou 121322 dia tou 7. Epei-d 322 ≡ 4 mod 6, èqoume 121322 ≡ 1214 mod7, all� 121 ≡ 2mod 7 kai �ra121322 ≡ 2mod 7.

2.4 Grammikèc IsotimÐec

'Estw α, β ∈ Z kai m ∈ N. JewroÔme to prìblhma Ôparxhc mh enìc akeraÐoux gia ton opoÐo na isqÔei

αx ≡ β mod m.(1)


H sqèsh (1) sthn opoÐa o x jewreÐtai wc �gnwstoc onom�zetai grammik isotimÐ-a. 'Opwc stic grammikèc Diofantikèc exis¸seic, k�je akèraioc x0 pou ikanopoieÐthn (1) lègetai lÔsh h rÐza thc grammik c isotimÐac (1). An x0 eÐnai mia lÔshthn (1), tìte gia k�je akèraio k èqoume

α(x0 + km) ≡ αx0 ≡ β mod m.

Sunep¸c ìloi oi akèraioi sthn kl�sh upoloÐpwn modm pou antiproswpeÔetaiapì ton x0 ikanopoioÔn thn (1). Apì t¸ra kai sto ex c ìtan lème lÔsh thc iso-timÐac (1) ja ennooÔme mia kl�sh upoloÐpwn mod m tètoia ¸ste k�je stoiqeÐox thc kl�shc ikanopoieÐ thn isotimÐa. To pl joc twn diaforetik¸n lÔsewn thc(1) sunep¸c ja eÐnai oi mh isotimÐec mod m lÔseic thc (1), dhlad to pl joctwn akèraiwn pou an koun se èna pl rec sÔsthma kl�sewn upoloÐpwn mod m

oi opoÐoi eÐnai lÔseic thc (1). Gia par�deigma, èstw h grammik isotimÐa

5x ≡ 3 mod 7.

'Ena pl rec sÔsthma kl�sewn upoloÐpwn mod 7 eÐnai to {0, 1, 2, 3, 4, 5, 6}. Do-kim�zontac autoÔc touc akèraiouc brÐskoume ìti o mìnoc akèraioc pou ikanopoieÐthn en lìgw isotimÐa eÐnai o x = 2. Sunep¸c aut h isotimÐa èqei mìno mÐa lÔsh,thn x ≡ 2 mod 7.

H isotimÐa (1) eÐnai isodÔnamh me th Diofantik exÐswsh αx −my = β. OilÔseic aut c perigr�fontai sto Je¸rhma 1.1.23: (α,m) - β den up�rqoun lÔseic,en¸ an (α, m)|β tìte up�rqoun �peirec lÔseic pou dÐnontai apì tic sqèseic

x = x0 +−m

(α, m)t, y = y0 − α

(α, m)t, t ∈ Z,

ìpou (x0, y0) eÐnai mia sugkekrimènh lÔsh thc exÐswshc.Sunep¸c h (1) èqei lÔsh an kai mìnon an (α, m)|β. An (α, m)|β, dhlad an

h (1) èqei lÔsh, èstw thn x0 mod m, gia na kajorÐsoume tic �llec lÔseic thc,jewroÔme tic lÔseic thc antÐstoiqhc Diofantik c exÐswshc. 'Estw

kaix1 = x0 − m

(α, m)t1, y1 = y0 − α

(α, m)t1

x2 = x0 − m

(α, m)t2, y2 = y0 − α

(α, m)t2


t1, t2 ∈ Z, dÔo lÔseic thc Diofantik c. DeÐqnoume ìti x1 ≡ x2 mod m an kaimìnon an t1 ≡ t2 mod (α, m).

Pr�gmati, èqoume x1−x2 =m

(α, m)(t2−t1). An (α, m)|t2−t1 tìte profan¸c

x1 − x2 ≡ 0modm kai antÐstrofa, an m|x1 − x2, tìtet2 − t1(α, m)

∈ Z dhlad

t2 − t1 ≡ 0mod (α,m).Me �lla lìgia x1 6≡ x2 mod m an kai mìnon an t1 6≡ t2 mod (α, m). Epeid

up�rqoun (α, m) to pl joc kl�seic upoloÐpwn modulo (α, m), up�rqoun (α, m)

to pl joc mh-isìtimec an� dÔo lÔseic thc (1). An jewr soume to pl rec sÔ-sthma {0, 1, . . . , (α, m)− 1} antipros¸pwn kl�sewn upoloÐpwn modulo (α, m)

tìte mia epilog (antipros¸pwn) mh isìtimwn lÔsewn mod m thc (1) dÐnetai apìtouc arijmoÔc

x0, x0 − m

(α, m), x0 − 2

m

(α, m), . . . , x0 − ((α, m)− 1)

m

(α, m).

'Etsi èqoume apìdeixh to ex c.

2.4.1 Je¸rhma. H grammik isotimÐa αx ≡ β mod m èqei lÔsh an kai mìnonan (α, m)|β. An (α, m)|β, tìte to pl joc twn diakekrimènwn lÔsewn thc enlìgw isotimÐac eÐnai Ðso me (α, m). Autèc oi lÔseic antiproswpeÔontai modulo

m apì touc akèraiouc x0 − m

(α,m)t, ìpou x0 eÐnai mia sugkekrimènh lÔsh thc

isotimÐac kai t ∈ {0, 1, 2, . . . , ((α, m)− 1)}.

Mia �llh apìdeixh autoÔ tou jewr matoc (pou ousiastik� eÐnai h Ðdia me thnprohgoÔmenh) sthrÐzetai sth sqèsh pou up�rqei metaxÔ twn kl�sewn upoloÐpwnmod n kai aut¸n mod m gia m,n ∈ N me n|m:

2.4.2 L mma. 'Estw m,n ∈ N, n|m. A α ∈ Z tìte h kl�sh α mod n eÐnai hxènh ènwsh twn kl�sewn upoloÐpwn (α + kn)modm, k = 0, 1, . . . ,

(mn − 1

).

Apìdeixh. 'Ola ta stoiqeÐa thc kl�sewc α mod n eÐnai oi akèraioi thc morf cα + λn, λ ∈ Z. Jewr¸ntac ìla aut� ta stoiqeÐa mod m, èqoume ìti α + λ1n ≡(α + λ2n)modm an kai mìnon an λ1n ≡ λ2n mod m isodÔnama (λ1 − λ2)n ≡0modm pou shmaÐnei ìti m|(λ1 − λ2)n, dhlad km = kn

m

n= (λ1 − λ2)n


km

n= λ1 − λ2, dhlad λ1 ≡ λ2 mod

m

n. Me �lla lìgia, k�je akèraioc α + λn,

λ ∈ Z eÐnai isìtimoc mod m me ènan apì touc arijmoÔc α, α+n, α+2n, . . . , α+(m

n−1

)n kai ìpwc mìlic deÐxame ìla aut� an� dÔo eÐnai mh isìtima mod m.

2h apìdeixh tou 2.4.1. An h isotimÐa (1) èqei lÔsh tìte ìpwc sthn 1h apìdeixhprèpei (α, m)|β. 'Estw ìti (α, m)|β.

To sÔnolo{

0, 1, 2, . . . ,( m

(α, m)−1

)}eÐnai èna pl rec sÔsthma antipros¸-

pwn kl�sewn upoloÐpwn modm

(α, m). Epeid

( α

(α, m),

m

(α, m)

)= 1, to sÔnolo

{0,

α

(α, m), 2

α

(α,m), . . . ,

( m

(α, m)−1

) α

(α, m)

}eÐnai epÐshc èna pl rec sÔsthma

antipros¸pwn kl�sewn upoloÐpwn modm

(α, m). Sunep¸c ènac apì touc arij-

moÔc x0 sto sÔnolo {0, 1, . . . ,

(m

(α, m)− 1

)}

prèpei na ikanopoieÐ thnα

(α, m)x0 ≡ β

(α, m)mod

m

(α, m)(2)

kai �ra thn

αx0 ≡ β mod m.(3)

'Ara h (1) èqei lÔsh.Epiplèon, ìloi oi akèraioi thc morf c x0 + λ

m

(α,m), dhlad oi isìtimoi tou

x0 modm

(α,m), ikanopoioÔn thn (2) kai �ra kai thn (3). Apì to 2.4.2, jètontac

n =m

(α, m)autoÐ oi akèraioi diamerÐzontai se (α,m) =

mm

(α,m)

kl�seic upoloÐpwn

mod m kai autèc oi kl�seic antiproswpeÔontai apì touc akèraiouc x0, x0 +m

(α, m), x0+2

m

(α, m), . . . , x0+((α, m)−1)

m

(α, m) isodÔnama apì touc akèraiouc

x0, x0 − m

(α, m), . . . , x0 − ((α,m)− 1)

m

(α, m).

Shmei¸noume ìti mìno ènac akèraioc arijmìc x0 sto sÔnolo{

0, 1, . . . ,

( m

(α, m)−1

)}ikanopoieÐ thn (2) afoÔ o

α

(α,m)eÐnai antistrèyimoc mod

m

(α, m),


dhlad h (2) èqei mia monadik lÔsh modm

(α,m). Gia na broÔme aut th lÔsh,

apl� efarmìzoume ton EukleÐdeio algìrijmo brÐskontac κ kai λ ètsi ¸ste

α

(α,m)κ +

m

(α, m)λ = 1.

ParadeÐgmata.

1. Na lujeÐ h grammik isotimÐa

15x ≡ 3mod 35.

Epeid (15, 35) = 5 - 3 aut eÐnai adÔnath dhlad den èqei lÔseic.

2. Na brejoÔn ìlec oi lÔseic (an up�rqoun) thc

7x ≡ 3 mod 10.

Epeid (7, 10) = 1|3, aut èqei lÔsh kai m�lista monadik , afoÔ o 7 èqeiantÐstrofo mod 10, ton 3mod 10. H lÔsh eÐnai h 9mod 10.

3. Na brejoÔn ìlec oi lÔseic thc

42x ≡ 84 mod 63.

EÐnai (42, 63) = 21 kai epeid 21|84, aut èqei tic lÔseic x0, x0 +6321

, x0 +

2 · 3, x0 +3 · 3, . . . , x0 +20 · 3mod 63, ìpou x0 mod 3 eÐnai h monadik lÔshmod3 thc 2x ≡ 4mod 3, dhlad h 2mod 3.

T¸ra jewroÔme sust mata grammik¸n isotimi¸n.'Estw to sÔsthma

α1x ≡ β1 mod m1

α2x ≡ β2 mod m2

...

αnx ≡ βn mod mn.


'Otan lème ìti ènac akèraioc x0 eÐnai lÔsh autoÔ tou sust matoc ennooÔme ìtio x0 eÐnai lÔsh k�je grammik c isotimÐac tou sust matoc.

Epeid k�je isotimÐa sto prohgoÔmeno sÔsthma orÐzei mia kl�sh upoloÐpwn,oi lÔseic tou sust matoc eÐnai oi arijmoÐ thc tom c aut¸n twn kl�sewn. Sune-p¸c mporoÔme na antikatast soume opoiad pote apì tic grammikèc isotimÐec memia diaforetik isotimÐa pou orÐzei tic Ðdiec kl�seic upoloÐpwn. Gia par�deigma,èstw to sÔsthma

3x ≡ 7mod 23

15x ≡ 12mod 18.

H pr¸th isotimÐa èqei th monadik lÔsh 10mod 23 kai h deÔterh èqei tic lÔseic2mod 18, 8mod 18 kai 14mod 18.

Sunep¸c ènac akèraioc x0 eÐnai mia lÔsh tou sust matoc an kai mìnon an o x0

an kei sthn tom twn kl�sewn 10mod 23 kai 2mod 18 sthn tom twn kl�sewn10mod 23 kai 8 mod 18 sthn tom twn kl�sewn 10mod 23 kai 14mod 18. 'Arato sÔsthma èqei tic Ðdiec lÔseic me tic lÔseic twn tri¸n susthm�twn

x ≡ 10mod 23

x ≡ 2mod 18

}x ≡ 10mod 23

x ≡ 8mod 18

}kai

x ≡ 10mod 23

x ≡ 14mod 18

}.

T¸ra gia tètoiou eÐdouc sust mata isqÔei to ex c:

2.4.3 Je¸rhma. To sÔsthma

x ≡ α1 modm1

x ≡ α2 modm2

èqei lÔsh an kai mìnon an (m1,m2)|α1 − α2.

Epiplèon, an to sÔsthma èqei lÔseic tìte ìlec autèc eÐnai oi akèraioi pouan kou se mia (monadik ) kl�sh upoloÐpwn mod [m1,m2]. Dhlad an y ≡α1 mod m1, y = α2 mod m tìte z ≡ α1 mod m1, z ≡ α2 mod m2 an kai mìno any ≡ z mod [m1,m2].


Apìdeixh. An x eÐnai mia lÔsh tou sust matoc tìte α1 ≡ α2 mod (m1,m2),dhlad (m1,m2)|α1 − α2.

AntÐstrofa, èstw ìti (m1,m2)|α1 − α2. H pr¸th isotimÐa apoteleÐtai apìtouc akèraiouc thc morf c x = α1 + κm1, κ ∈ Z. Prèpei na deÐxoume ìtiup�rqei ènac κ tètoioc ¸ste α1 + κm1 ≡ α2 mod m2 isodÔnama κm1 ≡ (α2 −α1)modm2. All� apì to Je¸rhma 2.4.1 ènac toul�qiston tètoioc κ up�rqeiafoÔ (m1,m2)|α2 − α1.

'Estw t¸ra ìti to sÔsthma èqei lÔsh, opìte up�rqei ènac κ0 ètsi ¸steκ0m1 ≡ (α2 − α1)modm2. Autìc o κ0 ja prèpei, sÔmfwna me to 2.4.1 (2h

apìdeixh), na an kei se mia orismènh kl�sh upoloÐpwn modm2

(m1,m2). Sunep¸c

ìloi oi akèraioi x pou ikanopoioÔn to sÔsthma ja eÐnai thc morf c x = α1 +(κ0 + λ

m2

(m1,m2)

)m1 = α1 + κ0m1 + λ[m1,m2], λ ∈ Z, dhlad ja eÐnai oi

akèraioi pou apoteloÔn mia orismènh kl�sh upoloÐpwn mod [m1, m2]. AutìmporeÐ na apodeiqjeÐ kai wc ex c. 'Estw y mia lÔsh tou sust matoc. An z eÐnaiepÐshc mia lÔsh, tìte z ≡ α1 ≡ y mod m1 kai z ≡ α2 ≡ y mod m2. Sunep¸c[m1,m2]|z−y, afoÔ m1|z−y kai m2|z−y. AntÐstrofa, an [m1, m2]|z−y tìteh z eÐnai mia lÔsh tou sust matoc an h y eÐnai.

FereipeÐn, sto prohgoÔmeno par�deigma, gia to sÔsthma

x ≡ 10mod 23

x ≡ 2mod 18

}

èqoume apì thn pr¸th isotimÐa touc arijmoÔc x = 10 + κ · 23 kai prèpei nakajorÐsoume ekeÐna ta κ gia ta opoÐa isqÔei 10 + κ · 23 ≡ 2 mod 18 κ · 23 ≡−8mod 18. Dhlad κ ≡ 2 mod 18, �ra x = 10 + 46 = 56. 'Ara ìloi oi akèraioithc kl�shc 56mod (18·23) eÐnai lÔseic tou sust matoc (opìte kai tou arqikoÔ).Gia ta sust mata

x ≡ 10mod 23

x ≡ 8mod 18

}kai

x ≡ 10mod 23

x ≡ 14mod 18

}

brÐskoume antÐstoiqa wc lÔseic tic kl�seic 332mod (18·23) kai 194 mod (18·23).


Opìte oi lÔseic tou arqikoÔ sust matoc

3x ≡ 7 mod 23

15x ≡ 12mod 18

}

eÐnai oi kl�seic mod18 · 23 pou antiproswpeÔontai apì touc arijmoÔc 56, 332kai 194.

To prohgoÔmeno je¸rhma genikeÔetai wc ex c.

2.4.4 Je¸rhma. To sÔsthma

x ≡ α1 mod m1

x ≡ α2 mod m2

...

x ≡ αr mod mr.

èqei lÔsh an kai mìnon an (mi, mj)|αi − αj gia k�je i 6= j, 1 ≤ i, j ≤ r.An isqÔei autì kai x0 eÐnai mia lÔsh tou sust matoc, tìte ìlec oi lÔseic tousust matoc eÐnai oi akèraioi pou an koun sthn kl�sh x0 mod [m1,m2, . . . ,mr].

Apìdeixh. Epagwg sto r.

2.4.5 Je¸rhma (Kinèziko). 'Estw m1,m2, . . . ,mr an� dÔo sqetik� pr¸toifusikoÐ arijmoÐ. 'Estw M = m1m2 · · ·mr. An α1, α2, . . . , αr eÐnai akèraioiarijmoÐ tìte k�je akèraioc x pou an kei sthn kl�sh

(∗)(

α1β1M

m1+ α2β2

M

m2+ · · ·+ αrβr

M

mr

)mod M

eÐnai h lÔsh tou sust matoc

x ≡ α1 mod m1

x ≡ α2 mod m2

...

x ≡ αr mod mr.


ìpou β1, β2, . . . , βr eÐnai antÐstoiqa antiprìswpoi twn antÐstrofwn kl�sewn twnM

m1mod m1,

M

m2mod m2, . . . ,

M

mrmod mr.

Apìdeixh. Epeid (mi,mj) = 1|αi − αj , to sÔsthma èqei lÔseic, sÔmfwna me

to 2.4.4. Epeid βiM

mi≡ 1 mod mi, i = 1, . . . , r, k�je akèraioc pou an kei sthn

kl�sh (∗) eÐnai profan¸c lÔsh tou sust matoc.Shmei¸noume ìti ènac akèraioc pou an kei sthn kl�sh (∗) eÐnai o

α1

(M

m1

)ϕ(m1)

+ α2

(M

m2

)ϕ(m2)

+ · · ·+ αr

(M

mr

)ϕ(mr)

.

Mia sun�rthsh f : N→ N onom�zetai pollaplasiastik an gia k�je m, n ∈N me (m, n) = 1 isqÔei

f(mn) = f(m)f(n).

Tètoiec sunart seic ja melet soume se epìmeno Kef�laio.Wc mia Efarmog tou Kinèzikou Jewr matoc ja d¸soume t¸ra mia apìdeixh

tou ex c shmantikoÔ jewr matoc.

2.4.6 Je¸rhma. H sun�rthsh ϕ tou Euler eÐnai pollaplasiastik .

Apìdeixh. Gia k�je jetikì akèraio κ, ja sumbolÐzoume me Uκ to sÔnolo twnantistrèyimwn kl�sewn mod κ. Sunep¸c ϕ(κ) = |Uκ|. 'Estw m,n ∈ N, me(m,n) = 1. Gia na deÐxoume ìti h ϕ eÐnai pollaplasiastik prèpei na deÐxoumeìti

|Umn| = |Um| |Un|.

OrÐzoume thn apeikìnish

Ψ : Umn → Um × Un

α mod mn → (α mod m,α mod n)

ìpou Um × Un eÐnai to Kartesianì ginìmeno tou Um epÐ tou Um. An deÐxoumeìti h Ψ eÐnai 1− 1 kai epÐ, tìte h zhtoÔmenh sqèsh ja isqÔei.


'Estw (α mod m,β mod n) ∈ Um×Un. Apì to Kinèziko Je¸rhma to sÔsth-ma

x ≡ α modm

x ≡ β mod n

èqei mia monadik lÔsh γ mod mn, dhlad

γ mod mn → (γ mod m, γ mod n) = (α mod m,β mod n).

Opìte h Ψ eÐnai epÐ.T¸ra an Ψ(α1 mod mn) = Ψ(α2 modmn), dhlad (α1 mod m,α1 mod n) =

(α2 modm, α2 mod n), autì shmaÐnei, apì ton orismì twn diatetagmènwn zeuga-ri¸n, ìti α1 ≡ α2 mod m kai α1 ≡ α2 mod n. Dhlad m|α1−α2 kai n|α1−α2.Epeid (m,n) = 1, èpetai ìti mn|α1 − α2, dhlad α1 ≡ α2 mod mn. Opìte hΨ eÐnai 1− 1.

2.4.7 Pìrisma. 'Estw m = pκ11 pκ2

2 · · · pκss h an�lush se pr¸touc enìc fusikoÔ

arijmoÔ m ≥ 2 Tìte

i) Gia k�je m > 2, o ϕ(m) eÐnai �rtioc.

ii) IsqÔei

ϕ(m) = ms∏

i=1

(1− 1

pi

)

Apìdeixh. UpologÐzoume ton ϕ(pκ), ìpou p eÐnai ènac pr¸toc. Apì ton orismìthc ϕ, o ϕ(pκ) eÐnai to pl joc ìlwn twn akeraÐwn α, 1 ≤ α ≤ pκ tètoiwn ¸ste(α, pκ) = 1. All� (α, pκ) = 1 an kai mìnon an p - α, dhlad (α, pκ) = 1 an kaimìnon an o α den eÐnai èna pollapl�sio tou p. Ta pollapl�sia tou p pou eÐnaimetaxÔ tou 1 kai tou pκ eÐnai oi arijmoÐ

p, 2p, 3p, . . . , p2, . . . , (pκ−1 − 1)p, pκ−1p

pou to pl joc touc eÐnai pκ−1. 'Ara

ϕ(pκ) = pκ − pκ−1 = pκ−1(p− 1).


Apì to 2.4.6 èqoume

ϕ(m) = ϕ(pκ11 · · · pκs

s ) =s∏

i=1

ϕ(pκii ) =

s∏

i=1

pκi−1i (pi − 1).

Opìte oi i) kai ii) isqÔoun.

Parat rhsh. Qrhsimopoi¸ntac to 2.4.7 i) mporoÔme na deÐxoume to Je¸rhmatou EukleÐdh, gia thn Ôparxh �peirou pl jouc pr¸twn, wc ex c. 'Estw ìtiup�rqoun mìno peperasmèno pl joc pr¸toi p1, p2, . . . , ps. JewroÔme ton arijmìn = p1p2 · · · ps > 2. An 1 < α < n, tìte o α diaireÐtai apì k�poion pi kaisunep¸c (α, n) 6= 1. Autì shmaÐnei ìti ϕ(n) = 1 pou eÐnai �topo afoÔ o ϕ(n)

eÐnai �rtioc kaj¸c n > 2. To Ðdio prokÔptei qrhsimopoi¸ntac to 2.4.7 ii) wcex c. Gia k�je m èqoume

1 = ϕ(mn) = mns∏

i=1

(1− 1

pi

)

ìpou o∏ (

1− 1pi

)eÐnai ènac stajerìc arijmìc. Autì eÐnai �topo.

Kef�laio 3Poluwnumikè Isotim�e kaiPrwtarqikè R�ze 'Estw

f(x) = αnxn + αn−1x

n−1 + · · · + α1x+ α0, me αn 6= 0,(1)'Ena polu¸numo mia metablht x me akèraiou suntelestè . 'Ena tètoio po-lu¸numo ja to lème akèraio polu¸numo. To sÔnolo ìlwn twn akera�wn po-luwnÔmwn ja to sumbol�zoume me Z[x]. To suntelest αi ja ton lème i-ostìsuntelest tou f(x). Eidik�, tou suntelestè αn kai α0 ja tou lème megisto-b�jmio suntelest kai stajerì ìro, ant�stoiqa. Pollè forè e�nai skìpimo,sthn par�stash enì poluwnÔmou na paremb�loume ìrou th morf 0xi. Autìsun jw to efarmìzoume ìtan jèloume na sugkr�noume (w pro mia idiìthta)dÔo polu¸numaf(x) =

n∑

i=0

αixi, αn 6= 0, g(x) =

m∑

i=0

βixi, βm 6= 0ìpou m ≤ n, gr�fonta g(x) =

n∑

i=0βix

i me βi = 0 gia i = m+ 1, . . . , n.131

132 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze 3.1 LÔsei Poluwnumik¸n Isotimi¸n'Estw m ∈ N. 'Eqoume dei ìti to apotèlesma th ektèlesh twn pr�xewnth prìsjesh kai tou pollaplasiasmoÔ se isìtimou arijmoÔ modm e�naiisìtimoi arijmo� modm. Sunep¸ k�je algebrik èkfrash pou proèrqetai apìepanalambanìmenh ektèlesh aut¸n twn pr�xewn ep� isìtimwn arijm¸n modmd�dei p�li isìtimou arijmoÔ modm. Opìte an α ≡ βmodm, tìte, gia ènaf(x) ∈ Z[x], èqoumedhlad f(α) ≡ f(β)modm

f(α) − f(β) = mt, t ∈ Z.Gia par�deigma, an f(x) = x4−x+1 kaim = 7, tìte èqoume f(−2) ≡ f(5)mod 7afoÔ −2 ≡ 5mod 7. Sugkekrimèna, f(−2) = 19 ≡ 621 = f(5)mod 7 (fusik� toant�strofo den isqÔei, fereipe�n èqoume f(0) ≡ f(1) ≡ f(2) ≡ f(4) ≡ 0mod 7).'Ena apì ta kentrik� probl mata th jewr�a arijm¸n e�nai h eÔresh twnakera�wn α (an up�rqoun) gia tou opo�ou isqÔei(2) f(α) ≡ 0modm,dhlad f(α) = mβ, gia k�poio β ∈ Z, ìpou f(x) e�nai èna dosmèno akèraiopolu¸numo. Sth gl¸ssa twn exis¸sewn autì to prìblhma e�nai isodÔnamo meto prìblhma th lÔsh th Diofantik ex�swsh f(x) = myw pro (akèraiou ) x kai y isodÔnama me to prìblhma th lÔsh th (alge-brik ) isotim�a (3) f(x) ≡ 0modm.'Ena akèraio α gia ton opo�o isqÔei h (2) lègetai lÔsh th (3). An α e�nailÔsh th (3) kai α ≡ rmodm, gia r ∈ {0, 1, . . . ,m − 1}, tìte ìpw e�dame jaisqÔei kai f(r) ≡ 0modm. Sunep¸ gia na broÔme ti lÔsei mia isotim�a (3)arke� na periorisjoÔme sto sÔnolo {0, 1, . . . ,m−1} ( opoiod pote �llo pl re

3.1. LÔsei Poluwnumik¸n Isotimi¸n 133sÔsthma antipros¸pwn twn kl�sewn upolo�pwn modulo m) kai sumfwnoÔme najewroÔme ìle ti lÔsei α th (3) pou e�nai isìtime me èna r, 0 ≤ r ≤ m− 1,w mia lÔsh. 'Etsi sto ex ìtan lème ìti o α kai o α′ e�nai lÔsei th (3) jaennooÔme ìti e�nai mh isìtime lÔsei .Ep�sh ap� ti idiìthte twn isotimi¸n prokÔptei ìti anf(x) =

n∑

i=0

αixi kai g(x) =

n∑

i=0

βixie�nai dÔo akèraia polu¸numa twn opo�wn oi ant�stoiqoi i-osto� suntelestè e�naiisìtimoi, dhlad αi ≡ βi modm, i = 0, 1, . . . , n, tìte èqoume

f(x) − g(x) =n

∑

i=0

(αi − βi)xi =

n∑

i=0

mtixi = mt(x)ìpou t(x) =

n∑

i=0tix

i ∈ Z[x], ti =αi − βi

m. Ant�strofa, an f(x) − g(x) =

mh(x), gia k�poio h(x) ∈ Z[x], tìte o i-ostì suntelest tou f(x) − g(x)e�nai �so me mhi, ìpou hi e�nai o i-ostì suntelest tou h(x), dhlad isqÔeiαi ≡ βi modm.3.1.1 Orismì . Duo polu¸numa f(x), g(x) ∈ Z[x] lègontai isìtima modm,kai gr�foume f(x) ≡ g(x)modm, an o i-ostì suntelest tou f(x) e�naiisìtimo modm me ton i-ostì suntelest tou g(x), gia ìla ta i.Sunep¸ isqÔei f(x) ≡ g(x) an kai mìnon an up�rqei h(x) ∈ Z[x] tètoio ¸stef(x) = g(x) + mh(x). E�nai eÔkolo na deiqje� ìti h sqèsh isotim�a modmmetaxÔ twn akèraiwn poluwnÔmwn e�nai mia sqèsh isodunam�a kai k�je kl�shisodunam�a perièqei èna monadikì polu¸numo tou opo�ou ìloi oi suntelestè an koun sto sÔnolo {0, 1, . . . ,m − 1}. Dhlad an f(x) ∈ Z[x], tìte up�rqeièna monadikì polu¸numo r(x) =

n∑

i=0rix

i, ri ∈ {0, 1, . . . ,m− 1} tètoio ¸ste(4) f(x) ≡ r(x)modm.ParathroÔme ìti ìtan ta f(x) kai g(x) e�nai stajer� polu¸numa h prohgoÔmenhsqèsh isodunam�a sump�ptei me th sqèsh isotim�a modm ep� twn akera�wnarijm¸n.

134 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze An duo polu¸numa f(x) kai g(x) e�nai isìtima modm, tìte e�nai fanerì ìtigia k�je α ∈ Z isqÔeif(α) ≡ g(α)modm.Idia�tera h ex�swsh f(x) ≡ 0modm èqei ti �die lÔsei me thn ex�swsh isotim�a

g(x) ≡ 0modm. Ton�zoume ìmw ìti to ant�strofo den isqÔei. Dhlad an giaf(x), g(x) ∈ Z[x] isqÔei f(α) ≡ g(α)modm, gia k�je α ∈ Z, tìte den prokÔpteianagkastik� ìti ja isqÔei f(x) ≡ g(x)modm. Gia par�deigma, èstw f(x) = xpkai g(x) = x, ìpou p e�nai èna pr¸to arijmì . Tìte profan¸ to f(x) dene�nai isìtimo mod p me to g(x), dhlad f(x) 6≡ g(x)mod p, all� apì to MikrìJe¸rhma tou Fermat gnwr�zoume ìti f(α) = αp ≡ α = g(α)mod p, gia k�jeα ∈ Z.3.1.2 Orismì . 'Estw f(x) =

n∑

i=0αix

i ∈ Z[x] kai m ∈ N. O bajmì degm f(x) modulo m or�zetai o megalÔtero fusikì i gia ton opo�o αi 6≡0modm.Fereipe�n, o bajmì tou f(x) = 4x3 + 5x2 + 10x + 6 modulo 3 e�nai 3 en¸modulo 4 e�nai 2, dhlad deg3 f(x) = 3, en¸ deg4 f(x) = 2.Sto ex dÔo polu¸numa f(x) kai g(x) pou e�nai isìtima modulo m ja tajewroÔme w to “�dio” polu¸numo. Sunep¸ modulo m ìla ta polu¸numaan�gontai se polu¸numa th morf (5) r(x) =

n∑

i=0

αixi, 0 ≤ αi ≤ m− 1.Idia�tera, ìle oi exis¸sei isotimi¸n f(x) ≡ 0modm, f(x) ∈ Z[x], mporoÔnna anaqjoÔn se exis¸sei isotimi¸n r(x) ≡ 0modm, ìpou to r(x) e�nai ènapolu¸numo th morf (5).Parat rhsh. W mhdenikì polu¸numo modulom e�nai k�je polÔwnumo pou oisuntelestè tou diairoÔntai ìloi dia tou m. 'Ola aut� ta polu¸numa (pou e�naiisìtima modm) jewroÔntai w to “�dio” polu¸numo pou to taut�zoume me tonakèraio arijmì 0. Gia thn apofug sÔgqush , ìtan gr�foume f(x) ≡ 0modm

3.2. Arijmhtik PoluwnÔmwn 135ja ennooÔme thn ex�swsh isotim�a kai ìqi ìti to f(x) e�nai isìtimo me to mhdenikìpolu¸numo. Sthn per�ptwsh pou to f(x) e�nai mhdenikì polu¸numo modm tìteja anafèretai rht¸ lègonta ìti to f(x) e�nai “ek tautìthta ” mhdèn.3.2 Arijmhtik PoluwnÔmwnW �mesh sunèpeia tou OrismoÔ 3.1.1, gia ti kl�sei isotim�a twn poluw-nÔmwn modm isqÔoun an�loge idiìthte me autè pou isqÔoun gia ti kl�sei upolo�pwn modm. Dhlad an f1(x) = g1(x)modm kai f2(x) ≡ g2(x)modmtìte isqÔoun oi isotim�e h(x)f1(x) ≡ h(x)g1(x)modm, gia k�je h(x) ∈ Z[x]

f1(x) ± f2(x) ≡ (g1(x) ± g2(x))modm kaif1(x)g1(x) ≡ f2(x)g2(x)modm.Gia par�deigma, èqoume x− i ≡ (x+ (m− i))modm, i = 0, 1, . . . ,m− 1 opìteisqÔei

f(x) = x(x− 1)(x − 2) · · · (x−m+ 1) ≡g(x) = (x+m)(x+m− 1) · · · (x+ 1)modm.Shmei¸noume ìti f(α) ≡ g(α) ≡ 0modm, gia k�je α ∈ Z, all� to f(x) ( to

g(x)) den e�nai ek tautìthta mhdèn modm, afoÔ (gia par�deigma) o megisto-b�jmio suntelest tou e�nai 1. 'Opw ep�sh en¸ f(α) ≡ 2g(α)modm, giak�je α ∈ Z, e�nai f(x) 6≡ 2g(x)modm, gia k�je m > 1, afoÔ oi megistob�jmioisuntelestè twn f(x) kai 2g(x) den e�nai isìtimoi modm. Ax�zei ep�sh na pa-rathr soume ìti f(α) ≡ g(α) ≡ 0mod (m!), gia k�je α ∈ Z, afoÔ, ìpw èqoumedei, to ginìmeno m diadoqik¸n akèraiwn diaire�tai dia m! (autì d�nei mia �llhapìdeixh ìti f(α) ≡ g(α)modm, kaj¸ m|m!), all� e�nai f(x) 6≡ g(x)modm!,gia m ≥ 3. Pr�gmati, o suntelest tou xm−1 isoÔtai me −m(m− 1)

2gia to

f(x) kai me m(m+ 1)

2gia to g(x), e�nai de −m(m− 1)

26≡ m(m+ 1)

2mod (m!)

136 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze gia m ≥ 3, diìti diaforetik� ja e�qame m = κ(m−1)! m = κ·2·3·4 · · · (m−1)pou e�nai �topo, afoÔ m− 1 ∤ m.Gia to bajmì tou ajro�smato kai tou ginomènou dÔo poluwnÔmwn modmisqÔoun ta ex .kai degm(f(x) + g(x)) ≤ max{degm f(x),degm g(x)}degm(f(x)g(x)) ≤ degm f(x) + degm g(x)Gia par�deigma,kai deg10((2x3 + x+ 1) + (8x3 + x2 + 4) ≡ (x2 + x+ 5)mod 10) = 2 < 3

deg10((5x2 + x+ 1)(2x + 1) ≡ (7x2 + 3x+ 1)mod 10) = 2 < 1 + 2 = 3.3.2.1 Orismì . 'Estw f(x), g(x) ∈ Z[x]. Ja lème ìti to f(x) diaire�tai diatou g(x)modm ( ìti e�nai èna pollapl�sio tou g(x) modm ìti to g(x) e�naièna diairèth tou f(x) modm) an up�rqei h(x) ∈ Z[x] tètoio ¸ste

f(x) ≡ g(x)h(x)modm.Fereipe�n, to f(x) = 7x2+5x+4 diaire�tai dia tou (3x+1)mod 12, afoÔ up�rqeito polu¸numo 4x2 + 5x+ 4 tètoio ¸ste(3x+ 1)(4x2 + 5x+ 4) = 12x3 + 19x2 + 17x+ 4 ≡ (7x2 + 5x+ 4)mod 12.3.2.2 L mma. 'Estw α ∈ Z kai f(x) ∈ Z[x]. Tìte isqÔei

f(x) ≡ (x− α)g(x)modm, g(x) ∈ Z[x](*)an kai mìnon an f(α) ≡ 0modm.Apìdeixh. An isqÔei h (∗), tìte prafan¸ f(α) ≡ 0modm. 'Estw ìti f(α) ≡0modm. An f(x) = αnx

n + · · · + α0, αn 6≡ modm, tìte èqoumef(x) − f(α) =

n∑

i=1

αi(xi − αi)

= (x− α)

n∑

i=1

αi(xi−1 + xi−2α+ · · · + αi−1)

= (x− α)g(x),

3.3. Poluwnumikè Isotim�e modulo èna Pr¸to Arijmì 137ìpou g(x) = αnxn−1 + · · · e�nai akèraio polu¸numo (me degm g(x) = n − 1).Opìte f(x) ≡ (x− α)g(x)modm.Parat rhsh. E�nai gnwstì ( dh apì th deuterob�jmia ekpa�deush) ìti an�-logh aut th Eukle�deia dia�resh pou isqÔei gia tou akèraiou arijmoÔ isqÔei kai gia ta polu¸numa me pragmatikoÔ suntelestè . Sugkekrimèna, an

f(x) kai h(x) e�nai dÔo polu¸numa me pragmatikoÔ suntelestè tìte up�rqounmonadik� polu¸numa π(x) kai v(x) me deg v(x) < deg h(x) tètoia ¸stef(x) = h(x)π(x) + v(x).Ton�zoume ìmw ìti an to f(x) kai to h(x) e�nai akèraia polu¸numa tìte genik�ta π(x) kai v(x) den e�nai akèraia. Sthn apìdeixh tou prohgoÔmenou L mmato blèpoume ìti sthn eidik per�ptwsh pou to f(x) e�nai akèraio kai h(x) = x−α,

α ∈ Z, tìte to π(x)(= g(x)) kai to v(x)(= f(α)) e�nai akèraia polu¸numa.Par�deigma. 'Estw f(x) = x2−1 kaim = 12. Epeid 52 ≡ 1mod 12, èqoumex2−1 ≡ (x2−52)mod 12, dhlad x2−1 = (x−1)(x+1) ≡ (x−5)(x+5)mod 12.Sunep¸ f(1) ≡ f(−1) ≡ f(5) ≡ f(−5) ≡ 0mod 12. Dhlad to polu¸numof(x) = x2 − 1 èqei toul�qiston 4 mh-isìtime r�ze mod12 an kai o bajmì toudeg12 f(x) = 2. Autì sumba�nei epeid (5 − 1)(5 + 1) = 4 · 6 ≡ 0mod 12, all�4 6≡ 0mod 12 kai 6 6≡ 0mod 12. Apì to L mma tou Eukle�dh, gnwr�zoume ìtiautì den isqÔei an o m e�nai èna pr¸to arijmì . 'Opw ja doÔme amèsw t¸ra,e�nai autì o lìgo gia ton opo�on mporoÔn na exaqjoÔn epakrib apotelèsmataìtan jewroÔme ta polu¸numa modulo ènan pr¸to arijmì.3.3 Poluwnumikè Isotim�e modulo èna Pr¸toArijmì3.3.1 Je¸rhma. 'Estw p èna pr¸to arijmì kai

f(x) = αnxn + · · · + α0, αn 6≡ 0mod p

138 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze èna akèraio polu¸numo. An r1, r2, . . . , rκ e�nai κ mh-isìtime r�ze tou f(x)mod ptìtef(x) ≡ (x− r1)(x− r2) · · · (x− rκ)g(x)mod pìpou g(x) ∈ Z[x] me degp g(x) = n−κ kai o megistob�jmio ìro tou g(x) e�naio αn. Eidik� an n = κ tìte g(x) = αn.Apìdeixh. Efarmìzoume epagwg sto κ. Gia κ = 1, to je¸rhma isqÔei apì toprohgoÔmeno L mma. Upojètoume ìti to je¸rhma isqÔei gia pl jo mh isìtimwnriz¸n ≤ κ− 1. 'Etsi èqoumef(x) ≡ (x− r1) · · · (x− rκ−1)h(x)mod pìpou h(x) ∈ Z[x] me degp h(x) = n− κ+ 1 kai o megistob�jmio ìro tou h(x)e�nai o αn. Epeid rκ e�nai r�za tou f(x)mod p kai rκ 6≡ ri mod p, i = 1, . . . , κ−1,èqoumeme (rκ − r1) · · · (rκ − rκ−1)h(rκ) ≡ 0mod p,

(rκ − r1) · · · (rκ − rκ−1) 6≡ 0mod p.Sunep¸ apì to L mma tou Eukle�dh pa�rnoume h(rκ) ≡ 0mod p kai �ra, apìto prohgoÔmeno L mma, èqoumeh(x) ≡ (x− rκ)g(x)mod pìpou g(x) ∈ Z[x], degp g(x) = n − κ kai o megistob�jmio ìro tou g(x) e�naio αn.Ta grammik� polu¸numa x − ri, i = 1, . . . , κ, onom�zontai grammiko� par�-gonte mod p tou f(x) kai h par�stash f(x) ≡ (x− r1) · · · (x− rκ)g(x)mod ponom�zetai paragontopo�hsh ( an�lush) tou f(x) se grammikoÔ par�gonte

mod p.Parade�gmata.

3.3. Poluwnumikè Isotim�e modulo èna Pr¸to Arijmì 1391. JewroÔme to polu¸numo f(x) = x3−2x. E�nai fanerì ìti gia k�jem ∈ N,èqoume f(x) ≡ x(x2 − 2)modm, dhlad to 0 e�nai r�za tou f(x)modm.Gia m = 7, to f(x)mod 7 èqei r�ze to 0, to 3 kai to −3(≡ 4mod 7)kai sunep¸ sÔmfwna me to 3.3.1, èqoume f(x) ≡ x(x − 4)(x − 3)mod 7.Gia m = 8, to f(x)mod 8 èqei r�ze to 0 kai to 4. 'Eqoume de f(x) ≡(x − 4)(x2 + 4x + 14)mod 8 (ìpw apaite�tai kai apì to 3.3.2). All� tof(x)mod 8 den analÔetai stou grammikoÔ par�gonte x kai x−4 kaj¸ ja èprepe na èqoume f(x) ≡ x(x−4)(x−α)mod 8, dhlad to α ja èprepena e�nai r�za tou f(x)mod 8 kai �ra α ≡ 0mod 8 α ≡ 4mod 8. All�tìte ja e�qame f(x) ≡ x2(x − 4)mod 8 f(x) ≡ x(x − 4)2 mod8 poushma�nei 4 ≡ 0mod 8 2 ≡ 0mod 8 pou e�nai �topo.2. JewroÔme to polu¸numo f(x) = x3 − 2x+ 1. Autì to polu¸numo èqei to1 r�za modm gia k�je m ∈ N, dhlad

f(x) ≡ (x− 1)(x2 + x− 1)modm, m ∈ N.Aplo� upologismo� d�nounf(x) ≡ (x− 1)(x− 2)2 mod 5

f(x) ≡ (x− 1)(x− 3)(x− 7)mod 11

f(x) ≡ (x− 1)(x− 4)(x− 14)mod 19.en¸ gia m = 3, 7, 13, 17 to polu¸numo x2 + x − 1 den èqei r�ze modm.('Ara autì to polu¸numo e�nai an�gwgo modm, m = 3, 7, 13, 17, dhlad den mpore� na ekfrasje� w ginìmeno dÔo �llwn akèraiwn poluwnÔmwnmodm bajm¸n < 2, afoÔ k�je paragontopo�hs tou ja èprepe na peri-lamb�nei ènan grammikì par�gonta).Qrhsimopoi¸nta to Je¸rhma 3.3.1 mporoÔme na apode�xoume m� ènan diafo-retikì trìpo th mia kateÔjunsh tou Jewr mato tou Wilson.3.3.2 Pìrisma (Je¸rhma Wilson). Gia k�je pr¸to p isqÔei

(p− 1)! + 1 ≡ 0mod p.

140 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze Apìdeixh. Apì to Je¸rhma tou Fermat h ex�swsh xp−1 − 1 ≡ 0mod p èqeilÔsei ti 1, 2, . . . , p − 1 mod p. Apì to 3.3.1, prokÔptei ìti xp−1 − 1 ≡ (x −1)(x− 2) · · · (x− (p− 1))mod p. Jètonta x = p, pa�rnoume −1 ≡ (p− 1)(p −2) · · · (p− (p − 1)) = ((p − 1)!)mod p.Parat rhsh. An sugkr�noume tou suntelestè tou xp−2, xp−3, . . . , x twnpoluwnÔmwn xp−1 − 1 kai (x− 1)(x− 2) · · · (x− p+ 1) kai an o p e�nai perittì pr¸to kai Sℓ, 1 ≤ ℓ < p − 1, e�nai to �jroisma twn ginomènwn ℓ diaforetik¸narijm¸n tou sunìlou {1, 2, . . . , p− 1} tìte Sℓ ≡ 0mod p.Apì to 3.3.1 ep�sh prokÔptei to ex basikì apotèlesma.3.3.3 Je¸rhma (Lagrange). K�je polu¸numo f(x) ∈ Z[x], bajmoÔ degp f(x) =

n èqei to polÔ n mh isìtime r�ze , gia k�je pr¸to arijmì p.Apìdeixh. 'Estw ìti to f(x) e�qe n+ 1 mh isìtime r�ze r1, r2, . . . , rn+1. Tìteapì to 3.3.1 prokÔptei ìtif(x) ≡ αn(x− r1) · · · (x− rn)mod p.All� apì thn upìjesh ep�sh èqoume

f(rn+1) ≡ 0 ≡ αn(rn+1 − r1) · · · (rn+1 − rn)mod p.Epeid ìmw αn 6≡ 0mod p, apì to L mma tou Eukle�dh, prèpei na up�rqeik�poio i = 1, 2, . . . , n ètsi ¸ste rn+1 − ri ≡ 0mod p pou e�nai �topo.3.3.4 Pìrisma. 'Estw f(x) ∈ Z[x]. Upojètoume ìti gia k�poio pr¸to p tof(x)mod p èqei perissìtere mh isìtime r�ze mod p apì ìse e�nai o bajmì deg f(x). Tìte to f(x) e�nai ek tautìthta mhdèn mod p.Apìdeixh. 'Estw f(x) = αnx

n + · · · + α0, αi ∈ Z. Upojètoume ìti gia k�poioi = 0, . . . , n, èqoume αi 6≡ 0mod p. 'Estw κ o megalÔtero de�kth gia ton opo�oακ 6≡ 0mod p. Tìte κ ≤ n kai to polu¸numo αnx

n + · · · + ακ+1xκ+1 e�nai ektautìthta mhdèn mod p (dhl. p|αi, i = κ+ 1, . . . , n). Sunep¸ to polu¸numo

(ακxκ + · · · + α0)mod p

3.3. Poluwnumikè Isotim�e modulo èna Pr¸to Arijmì 141èqei perissìtere apì κ mh isìtime r�ze mod p. Autì den mpore� na isqÔeisÔmfwna me to 3.3.3. 'Ara, gia k�je i = 0, . . . , n, prèpei p|αi.3.3.5 Je¸rhma. 'Estw f(x), g(x) ∈ Z[x] kai p èna pr¸to arijmì . An tof(x) diaire�tai dia tou g(x)mod p kai to f(x)mod p èqei tìse mh-isìtime r�ze ìse e�nai o bajmì degp f(x), tìte to �dio isqÔei kai gia to g(x)mod p.Apìdeixh. Apì thn upìjesh èqoume f(x) ≡ g(x)h(x)mod p gia k�poio h(x) ∈Z[x]. 'Estw

g(x) = βmxm + · · · + β0 kai h(x) = γℓx

ℓ + · · · + γ0me βm 6≡ 0mod p kai γℓ 6≡ 0mod p. Tìte èqoumef(x) ≡ (βmγℓx

m+ℓ + · · · + β0γ0)mod p, me βmγℓ 6≡ 0mod pdhlad degp f(x) = m+ ℓ. An α e�nai mia r�za tou f(x)mod p, dhlad f(α) ≡g(α)h(α) ≡ 0mod p, tìte apì to L mma tou Eukle�dh g(α) ≡ 0mod p h(α) ≡ 0mod p. Me �lla lìgia, k�je r�za tou f(x)mod p e�nai r�za toul�-qiston enì apì ta g(x)mod p kai h(x)mod p. Ant�strofa, k�je r�za enì apìta g(x)mod p kai h(x)mod p ja e�nai r�za tou f(x)mod p.T¸ra an to pl jo twn mh isìtimwn riz¸n tou g(x)mod p tou h(x)mod p tan mikrìtero tou m kai tou ℓ ant�stoiqa tìte to pl jo twn mh isìtimwnriz¸n tou f(x)mod p ja tan mikrìtero tou m+ ℓ = degp f(x) pou e�nai �toposÔmfwna me thn upìjesh. Sunep¸ to g(x)mod p kai to h(x)mod p prèpei naèqoun m kai ℓ mh isìtime r�ze ant�stoiqa.3.3.6 Pìrisma (Gen�keush tou Jewr mato tou Fermat). 'Estw p èna pr¸-to arijmì kai n èna diairèth tou p− 1. Tìte to polu¸numo (xn − 1)mod pèqei akrib¸ n r�ze .Apìdeixh. 'Estw p− 1 = nκ, opìte èqoume

xp−1 − 1 = (xn)κ − 1 = (xn − 1)(xn(κ−1) + · · · + 1).

142 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze Apì to Je¸rhma tou Fermat to (xp−1−1)mod p èqei akrib¸ ti r�ze 1, 2, . . . , p−1 pou e�nai p−1 to pl jo opìte kai to (xn−1)mod p èqei akrib¸ n mh-isìtime r�ze afoÔ degp(x

n − 1) = n kai (xp−1 − 1) ≡ (xn − 1)h(x)mod p.Suqn� ìtan jèloume na broÔme ti r�ze enì poluwnÔmou f(x)mod p, ppr¸to , metasqhmat�zoume to f(x)mod p se èna �llo polu¸numo g(x)mod ppou èqei akrib¸ ti �die mh-isìtime r�ze mod p me to f(x)mod p. Autìepitugq�netai efarmìzonta to Je¸rhma tou Fermat.3.3.7 Je¸rhma. 'Estw f(x) ∈ Z[x] kai p èna pr¸to arijmì . Tìte up�rqeièna polu¸numo g(x) ∈ Z[x] bajmoÔ deg g(x) < p tètoio ¸ste f(x) ≡ g(α)mod pgia k�je α ∈ Z.Apìdeixh. MporoÔme na upojèsoume ìti to f(x)mod p den e�nai stajerì. 'Estwdegp f(x) = n kai f(x) ≡ (αnx

n + · · ·+α0)mod p. An i ≥ 1 e�nai èna fusikì arijmì , tìte up�rqoun πi, ri ∈ Z tètoioi ¸ste i = (p−1)πi+ri me 1 ≤ r ≤ p−1.Pr�gmati, to sÔnolo S = {i− (p− 1)s ≥ 1/s ∈ Z} den e�nai kenì (afoÔ i ∈ S).Sunep¸ to S èqei el�qisto stoiqe�o ri = i− (p− 1)πi pou e�nai ≤ p− 1, diìtian tan > p − 1 tìte ja e�qame ri − (p − 1) = i − (p − 1)(πi + 1) ≥ 1 dhlad ri − (p − 1) ∈ S kai ri − (p− 1) < ri pou e�nai �topo.Gr�foume t¸ra k�je ìro αix

i tou f(x)mod p w αi(xp−1)πixri . An p ∤ α,

α ∈ Z, tìte αiαi ≡ αiα

ri mod p, lìgw tou Jewr mato tou Fermat. An p|α,α ∈ Z, tìte αiα

i ≡ 0 ≡ αiαri mod p.Sunep¸ to polu¸numo g(x) = αnx

rn + · · · + α0 ikanopoie� thn isodunam�af(α) ≡ g(α)mod p gia k�je α ∈ Z kai deg g(x) < p.Parathr sei .1. To prohgoÔmeno je¸rhma ja mporoÔse na apodeiqje� qrhsimopoi¸nta thn Eukle�deia dia�resh twn poluwnÔmwn me akèraiou suntelestè : An

ϕ1(x), ϕ2(x) ∈ Z[x] ìpou to ϕ2(x) èqei megistob�jmio suntelest 1 tì-te up�rqoun π(x), r(x) ∈ Z[x] me deg v(x) < ϕ2(x) v(x) = 0 tètoia¸ste ϕ1(x) = ϕ2(x)π(x) + v(x). Pr�gmati, an degϕ1(x) < degϕ2(x)

3.3. Poluwnumikè Isotim�e modulo èna Pr¸to Arijmì 143èqoume π(x) = 0 kai v(x) = ϕ1(x). An degϕ2(x) = 0 tìte ϕ2(x) = 1kai π(x) = ϕ1(x), v(x) = 0. An degϕ1(x) = 1 kai degϕ2(x) = 1 tìteϕ1(x) = α1x+α0 kai ϕ2(x) = x+β. 'Ara ϕ1(x) = α1ϕ2(x)+(α0−α1β).Opìte π(x) = α1 kai v(x) = α0 − α1β. Efarmìzoume t¸ra epagw-g sto bajmì tou ϕ1(x). Upojètoume ìti o isqurismì isqÔei gia k�-je ϕ1(x) ∈ Z[x] me degϕ1(x) ≤ n − 1. 'Estw èna ϕ1(x) ∈ Z[x] medegϕ1(x) = n, ϕ1(x) = αnx

n + · · · + α0 kai degϕ2(x) = m ≤ n (diafo-retik� e�dame ìti isqÔei). Ta polu¸numa ϕ1(x) kai αnxn−mϕ2(x) èqounto �dio megistob�jmio suntelest , ton αn. To polu¸numo

ϕ3(x) = ϕ1(x) − αnxn−mϕ2(x)èqei bajmì ≤ n − 1. 'Ara epagwgik� up�rqoun π′(x), v′(x) ∈ Z[x] me

v′(x) = 0 deg v′(x) < n tètoia ¸ste ϕ3(x) = ϕ2(x)π′(x)+v′(x). Opìteèqoume

ϕ1(x) = ϕ3(x) + αnxn−mϕ2(x)

= ϕ2(x)(π′(x) + αnx

n−m) + v′(x)

= ϕ2(x)π(x) + v(x)ìpou π(x) = π′(x) + αnxn−m, v(x) = v′(x) ∈ Z[x] kai ikanopoioÔn autìpou zhte�tai.Efarmìzonta aut th dia�resh sto ϕ1(x) = f(x) kai ϕ2 = xp−x, èqoume

f(x) = (xp − x)π(x) + v(x), π(x), v(x) ∈ Z[x], v(x) = 0 deg v(x) < p.'Ara telik� f(α) = v(α)mod p lìgw tou Jewr mato tou Fermat, mev(x) = 0 deg v(x) < p.2. Gia to polu¸numo g(x) pou anafèretai sto prohgoÔmeno je¸rhma mpo-roÔme na upojèsoume ìti èqei megistob�jmio suntelest 1. Pr�gmati,an g(x) = βmx

m + · · · + β0, βm 6≡ 0mod p tìte up�rqei γ ∈ Z meβmγ ≡ 1mod p kai to polu¸numo g′(x) = γg(x) ikanopoie� thn isodu-nam�a f(x) ≡ g′(α)mod p gia k�je α ∈ Z.

144 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze 3. Gia na kajor�soume an èna polu¸numo f(x)mod p, p pr¸to , bajmoÔn, 0 < n < p, èqei akrib¸ n mh isìtime r�ze , pollaplasi�zoume tof(x)mod p ep� ton ant�strofo mod p tou megistob�jmiou suntelest tou. 'Etsi pa�rnoume èna polu¸numo g(x)mod p bajmoÔ n me megisto-b�jmio suntelest 1. Opìte diair¸nta to xp − x dia tou g(x)mod pèqoume xp − x = g(x)π(x) + v(x), v(x) = 0 deg v(x) < n. Sunep¸ to g(x)mod p èqei n mh-isìtime r�ze mod p an kai mìnon an to v(x)e�nai ek tautìthta mhdèn mod p. Pr�gmati, an v(x) e�nai ek tautìthta mhdèn mod p tìte xp − x ≡ g(x)π(x)mod p, opìte apì to 3.3.5 to g(x)èqei akrib¸ n mh-isìtime mod p r�ze afoÔ to xp − x mod p èqei p mhisìtime r�ze . Ant�strofa, an to g(x)mod p èqei n mh-isìtime r�ze , tov(x) = ((xp − x) − g(x)π(x))mod p èqei toul�qiston n mh-isìtime r�ze afoÔ to (xp − x)mod p èqei p mh-isìtime r�ze . All� deg v(x) < n v(x) = 0 kai apì to Pìrisma 3.3.4 prèpei v(x) ≡ 0mod p ek tautìthta .Parade�gmata.1. 'Estw f(x) = x72 + x41 + x2 + 3. Tìte to f(x)mod 7 metasqhmat�zetaisto g(x) = (x6 +x5 +x2 +3)mod 7, afoÔ 72 = 6 ·11+6 kai 41 = 6 ·6+5.En¸ to f(x)mod 3 metasqhmat�zetai sto g(x) = (2x2 + x)mod 3, afoÔ72 = 2 ·35+2 kai 41 = 2 ·20+1, opìte f(x) = (x2)35x2 +(x2)20x+x2 +3kai g(x) = x2 + x+ x2 + 3 ≡ (2x2 + x)mod 3.2. 'Estw f(x) = x5 + 2x4 + 3x3 − 2x2 − x kai p = 5. To f(x)mod 5metasqhmat�zetai sto g(x) = x+2x4 +3x3−2x2−x = 2x4 +3x3−2x2 =

x2(2x2 +3x−2). Oi mh-isìtime r�ze tou g(x)mod 5 e�nai oi �die m� autè tou f(x)mod 5, pou eÔkola br�skoume ìti autè e�nai α1 ≡ 0 kai α2 ≡ 3.Eidikìtera èqoumekai f(x) ≡ x(x− 3)(x3 + 3x+ 2)mod 5

g(x) ≡ x2(x− 3)2 mod5.To polu¸numo (x3 +3x+2)mod 5 den èqei kam�a r�za mod 5 kai �ra e�naian�gwgo (afoÔ diaforetik� toul�qiston èna apì tou pijanoÔ par�gon-

3.4. Poluwnumikè Isotim�e modulo èna SÔnjeto Arijmì 145te ja èprepe na e�nai grammikì pou ja s maine ìti ja e�qe r�za).ParathroÔme ed¸ ìti an kai f(α) ≡ g(α)mod 5 gia k�je α ∈ Z, tof(x)mod 5 èqei dÔo r�ze pollaplìthta èna en¸ to g(x)mod 5 èqei dÔor�ze pollaplìthta dÔo.3. 'Estw f(x) = 5x4 − x3 + 4x2 + 4x + 3 kai p = 11. JewroÔme to g(x) ≡9f(x) ≡ (x4 − 9x3 + 3x2 + 3x+ 5)mod 11, ìpou 5 · 9 ≡ 1mod 11. Epeid x11 −x ≡ [(x4 − 9x3 +3x2 +3x+5)(x7 − 9x6 −x5 −x4 − 2x3 +3x+5)+

(2x3 − 2x2 − 9x− 3)]mod 11, to f(x) den èqei 4 mh-isìtime r�ze mod11afoÔ to (2x3 − 2x2 − 9x− 3)mod 11 den e�nai ek tautìthta mhdèn.4. To polu¸numo (xp + β)mod p èqei mìno mia r�za mod p afoÔ autì meta-sqhmat�zetai sto (x+ β)mod p (kaj¸ p = (p − 1) + 1 kai xp = xp−1x)kai e�nai αp + β ≡ (α + β)mod p, gia k�je α ∈ Z. Dhlad aut h r�zae�nai h x ≡ −βmod p.3.4 Poluwnumikè Isotim�e modulo èna SÔnje-to Arijmì'Estw m = pm11 · · · pms

s h an�lush se pr¸tou arijmoÔ enì fusikoÔ arijmoÔ mkai f(x) ∈ Z[x]. S� aut thn par�grafo ja doÔme pw mporoÔme na broÔme ti lÔsei th poluwnumik isotim�a f(x) ≡ 0modm ìtan gnwr�zoume ti lÔsei twn poluwnumik¸n exis¸sewn f(x) ≡ 0mod pi, i = 1, 2, . . . , s.Gia ènan fusikì arijmì n, ja sumbol�zoume me Λf (n) èna sÔnolo antipro-s¸pwn mh isìtimwn lÔsewn modn th f(x) ≡ 0modn kai me λf (n) o pl jo aut¸n, dhlad λf (n) = |Λf (n)|.3.4.1 Je¸rhma. H ex�swsh f(x) ≡ 0modm èqei lÔsh an kai mìnon an oiexis¸sei f(x) ≡ 0mod pmii , i = 1, 2, . . . , s èqoun lÔsei kai oi lÔsei aut¸nkajor�zoun ti lÔsei th f(x) ≡ 0modm. IsqÔei de

λf (m) = λf (pm11 ) · · · λf (pms

s ).

146 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze Apìdeixh. 'Estw α ∈ Λf (m). Epeid f(α) ≡ 0modm, profan¸ ja èqoumekai f(α) ≡ 0mod pmii , i = 1, . . . , s. Opìte gia k�je i = 1, . . . , s, up�rqei

αi ∈ Λf (pmii ) me α ≡ αi mod pmi

i . Dhlad èqoume mia apeikìnishΨ : Λf (m) → Λf (pm1

1 ) × · · · × Λf (pmss )me ψ(α) = (α1, . . . , αs), α ≡ αi mod pmi

i , i = 1, . . . , s.De�qnoume ìti h Ψ e�nai 1 − 1 apeikìnish. Pr�gmati, an ψ(α) = ψ(β) giaα, β ∈ Λf (m) autì ja s maine ìti α ≡ βmod pmi

i , i = 1, . . . , s. Epeid ìmw (pmi

i , pmj

j ) = 1, i 6= j, i, j = 1, . . . , s ja èprepe m|α − β, dhlad α ≡ βmodmkai �ra α = β, apì ton orismì tou Λf (m).T¸ra de�qnoume ìti h Ψ e�nai ep� apeikìnish. 'Estw (α1, . . . , αs) ∈ Λf (pm11 )×

· · · × Λf (pmss ). Apì to Kinèziko Je¸rhma up�rqei mia monadik lÔsh αmodmtou sust mato

x ≡ αi mod pmii , i = 1, . . . , s.Autì bèbaia den sunep�getai ìti ψ(α) = (α1, . . . , αs), all� ìmw pr�gmati,epeid

f(α) ≡ f(αi) ≡ 0mod pmii , i = 1, . . . , sprèpei kai f(α) ≡ 0modm (afoÔ (pmi

i , pmj

j ) = 1) kai �ra up�rqei k�poio β ∈Λf (m) me β ≡ αmodm ètsi ¸ste ψ(β) = (α1, . . . , αs). Sunep¸ h ψ e�nai ep�.Epeid h ψ e�nai 1 − 1 kai ep�, èqoume ìti

|Λf (m)| =

s∏

i=1

|Λf (pmii )| λf (m) =

s∏

i=1

λf (pmii ).E�nai de fanerì ìti h mèjodo th prohgoÔmenh apìdeixh d�nei mia mèjodokajorismoÔ twn lÔsewn th f(x) ≡ 0modm an gnwr�zoume ti lÔsei twn

f(x) ≡ 0mod pmii , i = 1, . . . , s.Par�deigma. Sto Par�deigma th §3.2 e�qame jewr sei to polu¸numo f(x) =

x2 − 1 kai m = 12 = 3 · 22. Oi lÔsei th f(x) ≡ 0mod 3 e�nai oi 1mod 3 kai−1mod 3 kai autè th f(x) ≡ 0mod 22 e�nai oi 1mod 4 kai −1mod4. 'Ara

3.4. Poluwnumikè Isotim�e modulo èna SÔnjeto Arijmì 147h f(x) ≡ 0mod 12 èqei 4 lÔsei kai autè kajor�zontai apì ti lÔsei twnsusthm�twn:x ≡ 1mod 3

x ≡ 1mod 4

x ≡ 1mod 3

x ≡ −1mod 4

x ≡ −1mod 3

x ≡ 1mod 4

x ≡ −1mod3

x ≡ −1mod4.Apì to Kinèziko Je¸rhma ( diaforetik�) oi lÔsei twn susthm�twn aut¸ne�nai ant�stoiqa 1mod 12, 7mod 12, 5mod 12 kai 11mod 12. An jewr soumeta sÔnola Λf (12) = {1, 7, 5, 11}, Λf (3) = |[1, 2} kai Λf (4) = {1, 3} gia thnapeikìnish Ψ : Λf (12) → Λf (3) × Λf (4) èqoume ψ(1) = (1, 1), ψ(7) = (1, 3),ψ(5) = (2, 1) kai ψ(11) = (2, 3).Apì to Je¸rhma 3.4.1 blèpoume ìti h melèth twn exis¸sewn f(x) ≡ 0modmgia sÔnjetou arijmoÔ m an�getai sth melèth twn exis¸sewn f(x) ≡ 0mod pκgia pr¸tou bajmoÔ p kai κ ≥ 1.Ja doÔme t¸ra ìti ep�sh gia ton prosdiorismì twn lÔsewn th f(x) ≡mod pκ arke� na broÔme ti lÔsei th f(x) ≡ 0mod p. Epakrib¸ , autì pouja k�noume e�nai na perigr�youme mia diadikas�a b�sei th opo�a oi lÔsei th

f(x) ≡ 0mod pκ+1ja prosdior�zontai apì ti lÔsei th f(x) ≡ 0mod pκ.Sunep¸ , xekin¸nta apì ti lÔsei th f(x) ≡ 0mod p ja prosdior�zontai oilÔsei th f(x) ≡ 0mod p2, met� oi lÔsei th f(x) ≡ 0mod p3, k.o.k.Gia na diatup¸soume th diadikas�a aut qreiazìmaste ton tÔpo tou Taylorgia ta akèraia polu¸numa. 'Estw f(x) to akèraio polu¸numo f(x) = αnx

n +

· · · + α0. H tupik par�gwgo tou f(x) or�zetai w to polu¸numo f ′(x) =

nαnxn−1 + · · · + α1, h deÔterh par�gwgo f ′′(x) tou f(x) e�nai h par�gwgo tou f ′(x) kai genik� h i-ost par�gwgo f (i)(x) tou f(x) e�nai h par�gwgo tou f (i−1)(x). EÔkola prokÔptei ìti isqÔei

(∗) (αf(x) + g(x))′ = αf ′(x) + g′(x), α ∈ Z, f(x), g(x) ∈ Z[x].

148 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze 3.4.2 L mma (TÔpo tou Taylor). 'Estw f(x) = αnxn + · · ·+α0 èna akèraiopolu¸numo. Tìte ìla ta polu¸numa

f (i)

i!, i = 1, 2, . . . , ne�nai akèraia polu¸numa kai isqÔei o tÔpo

f(x+ y) = f(x) +f ′(x)

1!y +

f ′′(x)

2!y2 + · · · + f (n)(x)

n!yn.Apìdeixh. Lìgw th prohgoÔmenh sqèsh (∗), arke� na apode�xoume to l mmagia to polu¸numo f(x) = xn. S� aut thn per�ptwsh èqoume

f(x+ y) = (x+ y)n = xn +

(

n

1

)

xn−1y + · · · +(

n

i

)

xn−iyi + · · · + ynìpou ìloi oi suntelestè (ni

) e�nai akèraioi. IsqÔei de f (i)(x) = n(n−1) · · · (n−

i+ 1)xn−i, o ≤ i ≤ n, opìte f (i)(x)

i!=

(

ni

)

xn−i ∈ Z[x].Melet�me t¸ra th sqèsh metaxÔ th ex�swsh f(x) ≡ 0mod pκ+1(1)kai th ex�swsh f(x) ≡ 0mod pκ, κ ≥ 1.(2) An α e�nai mia lÔsh th (2) mporoÔme na upojèsoume ìti α ∈ {0, 1, . . . , pκ −

1}. Oi arijmo� α + tpκ, 0 ≤ t ≤ p − 1, e�nai ìloi isìtimoi mod pκ all�ìqi mod pκ+1. Ap� thn �llh meri�, an β e�nai mia lÔsh th (1) mporoÔme naupojèsoume ìti β ∈ {0, 1, . . . , pκ+1 − 1}, e�nai de f(β) = λpκ+1 kai sunep¸ f(β) = ξpr, r = 1, . . . , κ. Dhlad an β e�nai lÔsh th (1), tìte aut e�-nai lÔsh kai twn exis¸sewn f(x) ≡ 0mod pr, r = 1, . . . , κ kai idiaitèrw th (2). Diair¸nta to β dia pκ, tìte mporoÔme na gr�youme monadik� to β w β = pκt + α, 0 ≤ t ≤ p − 1, 0 ≤ α ≤ pκ − 1, opìte β ≡ αmod pκ kai�ra f(β) ≡ f(α) ≡ 0mod pκ, dhlad to α e�nai lÔsh th (2). Sunep¸ , anα, 0 ≤ α ≤ pκ − 1, e�nai mia lÔsh th (2), ja prèpei na broÔme gia poia t,0 ≤ t ≤ p− 1, oi arijmo� pκt+ α e�nai lÔsei th (1).

3.4. Poluwnumikè Isotim�e modulo èna SÔnjeto Arijmì 149Parade�gmata.1. 'Estw h ex�swsh f(x) = x3 − 2x+ 1 ≡ 0mod 52.JewroÔme thn ex�swsh x3 − 2x + 1 ≡ 0mod 5. EÔkola br�skoume ìtiaut èqei ti lÔsei 1 kai 2 modulo 5. 'Ara oi pijanè lÔsei th f(x) ≡0mod 52 ja e�nai th morf 5t + 1 5t + 2, gia t = 0, 1, 2, 3, 4. Giax = 5t + 1, pa�rnoume (5t + 1)3 − 2(5t + 1) + 1 = λ52 52(5t3 + 3t2) +

5t = λ52 t = 5(−5t3 − 3t2 + λ). 'Ara mìno gia t = 0 èqoume lÔsh(thn profan x = 1) th f(x) ≡ 0mod 52. Gia x = 5t + 2, pa�rnoume(5t + 2)3 − 2(5t + 2) + 1 = λ52 5t(5t2 + 6t + 2) + 1 = λ · 5, pou denikanopoie�tai gia kanèna t kai �ra h f(x) ≡ 0mod 52 den èqei kam�a lÔshth morf 5t+ 2.2. 'Estw f(x) = x2 +x+ 7, p = 3. H ex�swsh f(x) ≡ 0mod 3 èqei monadik lÔsh thn x = 1 modulo 3. Oi pijanè lÔsei th f(x) ≡ 0mod 32 e�nai th morf 3t+1, t = 0, 1, 2. Sunep¸ ja prèpei na èqoume (3t+1)2 +3t+1+

7 = λ32 t2 + t+ 1 = λ, gia k�poio λ ∈ Z. Autì isqÔei kai gia t = 0 kaigia t = 1 kai gia t = 2. 'Ara oi x = 1, x = 4, x = 7mod 32 e�nai lÔsei th f(x) ≡ 0mod 32. Gia x = 1, oi pijanè lÔsei th f(x) ≡ 0mod 33 e�naith morf 32t+1, t = 0, 1, 2. Ja prèpei ìmw na isqÔei 3(3t2+t)+1 = λ·3gia k�poio λ. Epeid profan¸ autì e�nai adÔnato, h f(x) ≡ 0mod 33 denèqei kam�a lÔsh th morf 32t + 1. Gia x = 4, oi pijanè ant�stoiqe lÔsei th f(x) ≡ 0mod 33 e�nai oi 32t + 4, t = 0, 1, 2, kai gia na e�naiautè lÔsei ja prèpei gia k�poio λ ∈ Z na èqoume 3t2 +3t+1 = λ. AutìisqÔei gia k�je t = 0, 1, 2, opìte oi x = 4, x = 13 kai x = 22mod 33e�nai lÔsei . Gia x = 7, eÔkola prokÔptei ìti den èqoume kam�a lÔsh th f(x) ≡ 0mod 33 th morf 32t+ 7, t = 0, 1, 2.T¸ra diatup¸noume kai apodeiknÔoume to je¸rhma pou kajor�zei th sunj khthn opo�a prèpei na plhre� to t gia na e�nai h pκt + α lÔsh th (1), ìtan h αe�nai lÔsh th (2).

150 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze 3.4.3 Je¸rhma (L mma tou Hensel). 'Estw α mia lÔsh th (2), ìpou 0 ≤α ≤ pκ − 1. Tìte,

i) An f ′(α) 6≡ 0mod p, up�rqei mia monadik lÔsh th (1) th morf pκt + αmod pκ+1, gia k�poio t = 0, 1, . . . , p − 1, ìpou f ′(x) e�nai h tu-pik par�gwgo tou f(x).

ii) An f ′(α) ≡ 0mod p, sthn per�ptwsh pou isqÔei f(α) ≡ 0mod pκ+1up�rqoun p lÔsei th (1) kai autè e�nai oi pκt + α modulo pκ+1, t =

0, 1, . . . , p − 1, en¸ sthn per�ptwsh pou isqÔei f(α) 6≡ 0mod pκ+1 denup�rqei kam�a lÔsh th (1).Apìdeixh. An β = pκt+ α, tìte apì to 3.4.2 èqoumef(β) = f(α+ pκt) = f(α) + pκtf ′(α) + (pκt)2

f ′′(α)

2!+ · · · + (pκt)n

f (n)(x)

n!ìpou n e�nai o bajmì tou f(x). T¸ra, kaj¸ to α e�nai lÔsh th (2), gia k�poioλ ∈ Z, ja èqoume f(α) = λpκ. 'Araìpou f(β) = pκ(λ+ tf ′(α)) + p2κN

N =t2f ′′(α)

2!+ pκt3κ f

(3)(α)

3!+ · · · + p(n−2)κtnκ f

(n)(α)

n!e�nai akèraio , afoÔ apì to L mma 3.4.2 oi arijmo� f ′′(α)

2!, . . . ,

f (n)(α)

n!e�naiakèraioi. Epiplèon, epeid κ ≥ 1, e�nai κ+ 1 ≤ 2κ kai �ra

f(β) ≡ pκ(λ+ tf ′(α))mod pκ+1.Sunep¸ èqoume f(β) ≡ 0mod pκ+1, dhlad to β e�nai lÔsh th (1), an kaimìnon an pκ(λ + tf ′(α)) ≡ 0mod pκ+1 pou aut h isotim�a e�nai isodÔnamh methn isotim�a(3) tf ′(α) + λ ≡ 0mod p tf ′(α) ≡ −f(α)

pκmod p,afoÔ λ = f(α)

pκ ∈ Z.

3.4. Poluwnumikè Isotim�e modulo èna SÔnjeto Arijmì 151H (3) d�nei th sunj kh pou prèpei na plhre� to t gia na èqei h (1) lÔsh th morf α+ pκt. Gnwr�zoume apì ti exis¸sei isotimi¸n pr¸tou bajmoÔ, ìtii) an f ′(α) 6≡ 0mod p tìte h ex�swsh (3) w pro t èqei lÔsh (pou e�naimonadik ) thn

t ≡ −(f ′(α))∗f(α)

pκmod pìpou (f ′(α))∗ mod p e�nai h ant�strofh kl�sh th f ′(α)mod p.

ii) an f ′(α) ≡ 0mod p, tìte h (3) ikanopoie�tai an kai mìnon an f(α) ≡0mod p. Sunep¸ an f(α) 6≡ 0mod p tìte h (1) den èqei kam�a lÔsh th morf α + pκt, en¸ an f(α) ≡ 0mod p tìte ìle oi p dunatè timè tou β = α + pκtpou pa�rnei gia t = 0, 1, . . . , p− 1, e�nai lÔsei th (1).3.4.4 Pìrisma. 'Estw f(x) ∈ Z[x] kai p èna pr¸to arijmì tètoio ¸steoi poluwnumikè isotim�e f(x) ≡ 0mod p kai f ′(x) ≡ 0mod p den èqoun kam�akoin lÔsh. Tìte to pl jo twn lÔsewn th f(x) ≡ 0mod p,isoÔtai me topl jo twn lÔsewn th f(x) ≡ 0mod p, gia k�je κ ∈ N. Eidik� an p ∤ n,tìte gia k�je α ∈ Z oi isotim�e xn ≡ αmod pκ kai xn ≡ αmod p èqoun to �diopl jo lÔsewn gia k�je κ ∈ N.Apìdeixh. 'Estw α, 0 ≤ α ≤ p − 1, mia lÔsh th f(x) ≡ 0mod p. Apì thnupìjesh prèpei f ′(α) 6≡ 0mod p kai sÔmfwna me 3.4.3, up�rqei monadik lÔshth f(x) ≡ 0mod p2 th morf β = α + tp, t = 0, . . . , p − 1. All� aut h lÔsh β e�nai kai lÔsh th f(x) ≡ 0mod p kai �ra prèpei f ′(β) 6≡ 0mod p.Sunep¸ prèpei na up�rqei monadik lÔsh th f(x) ≡ 0mod p3 th morf γ = β + tp2. Suneq�zonta m� autì ton trìpo, blèpoume ìti se k�je lÔsh th f(x) ≡ 0mod p antistoiqe� monadik lÔsh th f(x) ≡ 0mod pκ, gia k�je κ ∈ N.Sthn eidik per�ptwsh f(x) = xn − α, α ∈ Z, p ∤ n, èqoume f ′(x) = nxn−1kai �ra h f ′(x) ≡ 0mod p e�nai isodÔnamh me thn xn−1 ≡ 0mod p. Epeid hxn − α ≡ 0mod p, me α 6≡ 0mod p, den èqei lÔsh thn mhdenik , oi dÔo isotim�e xn − α ≡ 0mod p kai xn − α ≡ 0mod pκ èqoun to �dio pl jo lÔsewn. Anα ≡ 0mod p, tìte to pl jo twn lÔsewn kai twn dÔo e�nai 1, afoÔ h monadik lÔsh tou e�nai h mhdenik .

152 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze Par�deigma. 'Estw m = 1125 = 53 · 32. Na luje� h isotim�af(x) = 4x4 + 9x3 − 5x2 − 21x+ 61 ≡ 0modm.SÔmfwna me to Je¸rhma 3.4.1 oi lÔsei th isotim�a aut kajor�zontai apìti lÔsei twn isotimi¸n

f(x) ≡ 0mod 53 kai(a)f(x) ≡ 0mod 32.(b)Gia ton kajorismì twn lÔsewn th (a), br�skoume ti lÔsei th f(x) ≡ 0mod 5.Oi pijanè lÔsei aut e�nai 0, 1, 2, 3, 4mod 5 kai epeid f(2) = 135 ≡ 0mod 5,

f(3) = 520 ≡ 0mod 5, oi 2mod 5 kai 3mod 5 e�nai lÔsei th . E�nai de f(x) ≡2(x − 2)(x − 3)(2x2 − 3x + 8)mod 5 kai �ra to 2 kai to 3mod 5 e�nai oi mìne lÔsei .T¸ra br�skoume ti lÔsei th f(x) ≡ 0mod 52. Oi pijanè lÔsei aut e�nai oi 2 + 5t kai 3 + 5t, t = 0, 1, 2, 3, 4. Epeid f ′(x) = 16x3 + 27x2 −10x − 21, blèpoume ìti f ′(2) = 195 ≡ 0mod 5 kai f(2) = 135 6≡ 0mod 52.'Ara sÔmfwna me to Je¸rhma 3.4.3, kanèna apì tou arijmoÔ 2 + 5t e�nailÔsh th f(x) ≡ 0mod 52 kai �ra kai th f(x) ≡ 0mod 53. Ep�sh èqoumef ′(3) = 624 6≡ 0mod 5 kai �ra up�rqei mia monadik lÔsh th f(x) ≡ 0mod 52th morf 3 + 5t, t = 0, 1, 2, 3, 4. Br�skoume ìti gia t = 4, èqoume f(23) =

1225800 ≡ 0mod 52, dhlad to 23mod 52 e�nai lÔsh th f(x) ≡ 0mod 52.E�nai de f ′(23) = 208704 6≡ 0mod 5 kai �ra up�rqei mia monadik lÔsh th f(x) ≡ 0mod 53 th morf 23 + 52t, t = 0, 1, 2, 3, 4. EÔkola br�skoume ìtigia t = 2, èqoume f(73) ≡ 0mod 53, dhlad h 73mod 53 e�nai lÔsh th (a). Giati lÔsei th (b) br�skoume ti lÔsei th f(x) ≡ 0mod 3 pou e�nai oi 1 kai2mod 3 kai èqoume f(x) ≡ (x − 1)2(x − 2)2 mod3. Sunep¸ oi pijanè lÔsei th (b) e�nai th morf 1 + 3t kai 2 + 3t, t = 0, 1, 2. Epeid f ′(1) ≡ 0mod 3kai f(1) 6≡ 0mod 9, h (b) den èqei kam�a lÔsh th morf 1 + 3t. Ep�sh f ′(2) = 195 ≡ 0mod 3 kai f(2) = 135 ≡ 0mod 9. Sunep¸ sÔmfwna me to3.4.3 up�rqoun 3 lÔsei th (b) th morf 2+3t, oi 2, 5 kai 8mod 9. Telik� oi

3.4. Poluwnumikè Isotim�e modulo èna SÔnjeto Arijmì 153lÔsei th f(x) ≡ 0mod 1125 d�dontai apì ti lÔsei twn susthm�twn (Je¸rhma3.4.1)x ≡ 73mod 53

x ≡ 8mod 9

}

x ≡ 73mod 53

x ≡ 2mod 9

}

x ≡ 73mod 53

x ≡ 5mod 9pou e�nai ant�stoiqa 323mod 1125, −52mod 1125 kai −427mod 1125.Parat rhsh. An α e�nai mia r�za mod p tou f(x)mod p kai èqoume(*) f(x) ≡ (x− α)mg(x)mod p, me g(α) 6≡ 0mod ptìte o arijmì m onom�zetai pollaplìthta th αmod p.Apì ton tÔpo tou Taylor èqoumef(x) = f(α) + (x− α)

f ′(α)

1!+ (x− α)2

f ′′(α)

2!+ · · · + (x− α)n

f (n)(α)

n!ìpou n e�nai o bajmì tou f(x). Sunep¸ an α e�nai r�za tou f(x)mod p tìte(**) f(x) ≡(

(x− α)f ′(α)

1!+ (x− α)2

f ′′(α)

2!+ · · ·

)

mod p.An h pollaplìthta th α e�nai ≥ 2, tìte apì thn (∗) prokÔptei ìtif ′(x) ≡ (m(x− α)m−1g(x) + (x− α)mg′(x))mod pkai �ra f ′(α) ≡ 0mod p. Ep�sh , apì thn (∗∗), isqÔei kai to ant�strofo, kaj¸ an f ′(α) ≡ 0mod p tìte

f(x) ≡ (x− α)2[

f ′′(α)

2+ (x− α)

f ′′′(α)

3!+ · · ·

]

mod pkai �ra f(x) ≡ (x − α)mg(x)mod p gia k�poio m ≥ 2 kai g(x) 6≡ 0mod p.Sunep¸ mia r�za αmod p tou f(x)mod p èqei pollaplìthta ≥ 2 an kai mìnonan f ′(α) ≡ 0mod p. Ap� autì to gegonì sumpera�noume ìti h per�ptwsh ii) tou3.4.3 isqÔei mìno gia ti r�ze tou f(x)mod p me pollaplìthta ≥ 2. Autì d�nei�mesh apìdeixh tou Por�smato 3.4.4.

154 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze 3.5 Prwtarqikè R�ze S� aut thn par�grafo ja efarmìsoume th jewr�a twn poluwnumik¸n isotimi¸nmodulo èna pr¸to arijmì p kai ti idiìthte twn isotimi¸n dun�mewn mia kl�sh αmod p.'Estw m > 1 èna fusikì arijmì kai αmodm mia antistrèyimh kl�shmodm. Sto Kef�laio II sel. 37 or�soume thn t�xh th kl�sh αmodm w tomikrìtero jetikì akèraio ν gia ton opo�on isqÔei αν ≡ 1modm. Merikè forè thn t�xh th αmodm th sumbol�zoume me om(α) apl� o(α) ìtan e�nai gnwstììti anaferìmaste mìno sti kl�sei modm. (Shmei¸noume ìti palaiìtera e�qeepikrat sei h orolog�a: “o α an kei ston ekjèth ν modulo m”). Opoted poteei to ex anafèroume t�xh mia kl�sh αmodm ( enì akèraiou α modulo

m) èmesa ja ennooÔme ìti aut e�nai antistrèyimh kl�sh (afoÔ an (α,m) 6= 1kai αν ≡ 1modm tìte to αν−1 ja tan lÔsh th ex�swsh αx ≡ 1modm pouaut den èqei lÔsh kaj¸ o (α,m) den diaire� to 1).To epìmeno je¸rhma perilamb�nei merikè aplè all� basikè idiìthte pousqet�zontai me thn t�xh twn kl�sewn modulo m.3.5.1 Je¸rhma.i) IsqÔei κ ≡ λmod o(α) an kai mìnon an ακ ≡ αλ modm.ii) 'Ena akèraio κ diaire�tai apì thn t�xh o(α) an kai mìnon an ακ ≡

1modm. Idia�tera, p�nta isqÔei o(α)|ϕ(m).iii) 'Ole oi dun�mei 1 = α0, α, α2, . . . , αo(α)−1 e�nai an� dÔo mh isìtimoi akè-raioi modm kai e�nai lÔsei th ex�swsh xo(α) ≡ 1modm.iv) H t�xh mia dÔnamei tou α d�detai sunart sei th t�xh tou α w ex

o(ακ) =o(α)

(κ, o(α)).Idiaitèrw an o(α) = κλ tìte o(ακ) = λ.

v) An (o(α), o(β)) = 1 tìte o(αβ) = o(α)o(β).

3.5. Prwtarqikè R�ze 155vi) 'Estw αmodm kai βmodm dÔo antistrèyime kl�sei . Tìte up�rqei miakl�sh γmodm tètoia ¸ste o(γ) = [o(α), o(β)].vii) H megalÔterh t�xh ìlwn twn t�xewn twn kl�sewn modm diaire�tai apìthn t�xh k�je kl�sh modm.Apìdeixh. i) Autì èqei dh apodeiqje� sthn Parat rhsh met� to Pìrisma 2.3.15.

ii) Autì prokÔptei apì to i) jètonta λ = 0 kai κ = ϕ(m). Autì ep�sh èqei apodeiqje� sthn apìdeixh tou Euler tou jewr mato tou Euler (sel. 81).iii) Blèpe sel. 81.iv) Apì to ii) èqoume ìti ακλ = (ακ)λ ≡ 1modm an kai mìnon an κλ ≡

0mod o(α). H teleuta�a isotim�a e�nai isodÔnamh me thn κ

(κ, o(α))λ ≡ 0mod

(

o(α)

(κ, 0(α))

) kai aut me thnλ ≡ 0mod

o(α)

(κ, o(α))afoÔ (

κ

(κ, o(α)),

o(α)

(κ, o(α))

)

= 1.Epeid o mikrìtero jetikì akèraio λ gia ton opo�o isqÔei h teleuta�a isotim�ae�nai profan¸ o o(α)

(κ, o(α)), prokÔptei ìti o(ακ) =

o(α)

(κ, o(α)).

v) 'Estw o(α) = κ, o(β) = λ kai o(αβ) = µ. Tìte1 ≡ (αβ)κλ ≡ (αβ)µκ ≡ βµκ ≡ αµλ modm.Opìte κ|µλ kai λ|µκ. Epeid (κ, λ) = 1, apì to L mma tou Eukle�dh, prèpei κ|µkai λ|µ. 'Ara kai κλ|µ, p�li lìgw tou ìti (λ, κ) = 1. Epeid (αβ)κλ ≡ 1modmprèpei kai µ|κλ �ra telik� µ = κλ.

vi) 'Estw o(α) = κ kai o(β) = λ. Tìte up�rqoun κ′, λ′ ∈ N tètoioi ¸steκ′|κ, λ′|λ, (κ′, λ′) = 1 kai κ′ · λ′ = [κ, λ]. Gia par�deigma, an κ = pκ1

1 · · · pκss ,

λ = pλ11 · · · pλs

s e�nai h an�lush twn κ kai λ se pr¸tou par�gonte , mpo-roÔme na upojèsoume met� apì mia kat�llhlh ar�jmhsh ìti max(κ1, λ1) =

κ1, . . . ,max(κt, λt) = κt,max(κt+1, λt+1) = λt+1, . . . ,max(κs, λs) = λs. Opì-te oi arijmo� κ′ = pκ11 · · · pκt

t , λ′ = pλt+1

t+1 · · · pλss èqoun thn apaitoÔmenh idiìthta.T¸ra èqoume apì thn iv)

o(ακ/κ′

) =κ

(

κκ′ , κ

) = κ′ kai o(βλ/λ′

) = λ′.

156 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze Opìte apì thn v) jètonta γ = λκ/κ′

βλ/λ′ , pa�rnoume o(γ) = κ′λ′ = [κ, λ].vii) 'Estw ℓ h megalÔterh t�xh metaxÔ ìlwn twn t�xewn twn antistrèyimwnkl�sewn modm. 'Estw αmodm me o(α) = ℓ. Opìte gia k�je antistrèyimhkl�sh βmodm e�nai o(α) ≥ o(β). Apì to v) up�rqei γmodm tètoia ¸ste

o(γ) = [0(α), o(β)]. Sunep¸ ℓ ≤ o(γ) ≤ ℓ kai �ra ℓ = o(γ). Opìte o(β)|ℓ.Parathr sei .1. H t�xh o(α−1) th ant�strofh th αmodm isoÔtai me thn t�xh o(α) th αmodm. Dhlad isqÔei

o(α−1) = o(α).Pr�gmati, èstw o(α−1) = ν kai o(α) = κ. 'Eqoume (α−1)κ = 1(α−1)κ ≡ακ(α−1)κ = (αα−1)κ ≡ 1κ = 1modm, opìte apì thn ii) prèpei ν|κ.'Omoia èqoume αν = 1αν ≡ (α−1)ναν = (α−1α)ν ≡ 1modm kai �ra κ|ν,dhlad κ = ν. Autì to gegonì ma epitrèpei na jewroÔme sto 3.5.1 iv)kai arnhtikè dun�mei .2. H t�xh o(αβ) p�nta diaire� to el�qisto koinì pollapl�sio ε = [o(α), o(β)],afoÔ (αβ)ε = αεβε ≡ 1modm, kaj¸ o(α)|ε kai o(β)|ε. Genik� denisqÔei o(αβ) = [o(α), o(β)]. Gia par�deigma, an m = 11, tìte 2−1 ≡6mod 11, 3−1 ≡ 4mod 11, 5−1 ≡ 9mod 11, 7−1 ≡ 8mod 11, 10−1 ≡10mod 11 kai èqoume o(2) = o(6) = 10, o(3) = o(4) = o(5) = o(7) =

o(8) = o(9) = 5 kai o(10) = 2. E�nai de o(2 · 5) = 2 6= [o(2), o(5)] = 10,o(2 ·3) = 10 = [0(2), o(3)], o(9 ·10) = o(90) = o(2) = 10 = [o(9), o(10)] =

[5, 2] = 10, ìpw prèpei na isqÔei kai apì to v).3. H 3.5.1 v) isqÔei kai gia perissìtere apì dÔo kl�sei . Dhlad anα1 modm, . . . , αs modm e�nai s kl�sei pou oi t�xei tou an� dÔo e�-nai pr¸te metaxÔ tou tìte h t�xh th α1α2 · · ·αs modm isoÔtai me toginìmeno twn t�xewn twn αi modm.4. Kaj¸ oi t�xei twn kl�sewn modm diairoÔn to ϕ(m), gia thn eÔreshth t�xh mia kl�sh arke� na exet�zoume ti dun�mei th kl�sh me

3.5. Prwtarqikè R�ze 157ekjète pou diairoÔn to ϕ(m). Gia par�deigma, an m = 8, oi t�xei twn3mod 8, 5mod 8 kai 7mod 8 ja prèpei na e�nai 2 ϕ(8) = 4. E�nai de32 ≡ 52 ≡ 72 ≡ 1mod 8. S� autì to par�deigma blèpoume ìti e�naidunatìn na up�rqoun diairète tou ϕ(m) pou na mhn e�nai t�xei k�poia kl�sh modm.5. Mia qr simh idiìthta gia thn eÔresh th t�xh mia kl�sh αmodm, giaèna sÔnjeto arijmì m e�nai h ex . An m = m1m2, m1 6= 1, m2 6= 1, tìte

o[m1,m2](α) = [om1(α), om2(α)].Pr�gmati, èstw om1(α) = κ, om2(α) = λ, o[m1,m2](α) = µ kai ε =

[κ, λ]. IsqÔei αε ≡ 1modm1 kai αε ≡ 1modm2 kai sunep¸ αε ≡1mod [m1,m2]. 'Ara, apì to 3.5.1 ii), prèpei to µ na diaire� to ε. Ep�sh epeid αµ ≡ 1mod [m1,m2], èqoume kai αµ ≡ 1modm1, αµ ≡ 1modm2,opìte apì to 3.5.1 ii) ja prèpei to µ na e�nai pollapl�sio tou κ kai tou λ�ra kai tou ε. Sunep¸ ε = µ. Gia par�deigma, èstwm = 55 = 5·11. E�naio5(4) = 2, o11(4) = 5, opìte o55(4) = 10, ìpw ep�sh o5(28) = o5(3) = 4,o11(28) = o11(6) = 10 kai sunep¸ o55(28) = 20. 'Estw m = 32 · 5 = 45,tìte o32(7) = 3, o5(7) = o5(2) = 4 kai �ra o45(7) = 12. Shmei¸noume ìtih prohgoÔmenh idiìthta epagwgik� pa�rnei th morf

o[m1,m2,...,ms](α) = [om1(α), om2(α), . . . , oms(α)].Opìte an m = pλ11 · · · pλs

s e�nai h an�lush tou m se pr¸tou tìteom(α) = [o

pλ11

(α), . . . , opλs

s(α)].Sunep¸ to prìblhma eÔresh th t�xh om(α) an�getai sto prìblhma eÔre-sh th t�xh opn(α) ìpou p e�nai èna pr¸to arijmì . All� ìpw ja de�xoumet¸ra kai autì an�getai sthn eÔresh th t�xh op(α).1η per�ptwsh: p = 2. 'Estw α èna perittì akèraio . Tìte α ≡ 1mod 4

α ≡ 3 ≡ −1mod 4. Opìteo4(α) =

{

1 an α ≡ 1mod 4

2 an α ≡ −1mod4.

158 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze Jètoume κ = o4(α) kai λ = o2n(α). An α = 1 −1 tìte profan¸ λ = 1 2.Upojètoume ìti α 6= ±1. 'Estw 2µ|ακ − 1 all� 2µ+1 ∤ ακ − 1. Ja de�xoume ìtiλ =

{

κ an 2 ≤ n ≤ µ

2n−µκ an n ≥ µ.'Estw 2 ≤ n ≤ µ. Tìte, epeid ακ ≡ 1mod 2µ, isqÔei ακ ≡ 1mod 2n kaisunep¸ , apì to 3.5.1 ii), λ|κ. All� ep�sh αλ ≡ 1mod 2n opìte kai αλ ≡1mod 4 kai sunep¸ κ|λ. 'Ara κ = λ.'Estw n ≥ µ. Epeid 2µ+1 ∤ ακ − 1, o ακ gr�fetai w ακ = 1 + 2µt, ìpou te�nai perittì arijmì . Tìte gia n ≥ µ isqÔei kai

(ακ)2n−µ

= 1 + 2ntn, ìpou tn perittì .Pr�gmati, efarmìzoume epagwg sto n. Gia n = µ h sqèsh aut d�detai. 'Estwìti isqÔei gia n, n ≥ µ. Tìte(ακ)2

n+1−µ= (1 + 2ntn)2 = 1 + 2n+1tn + 22nt2n

= 1 + 2n+1(tn + 2n−1t2n) = 1 + 2n+1tn+1,ìpou tn+1 e�nai perittì , kaj¸ n ≥ µ ≥ 2.Ap� autì prokÔptei ìti λ = 2n−µκ, an n ≥ µ.2η per�ptwsh: p perittì . 'Estw κ = op(α), λ = opn(α) kai pµ|ακ − 1 all�pµ+1 ∤ ακ − 1. MporoÔme p�li na upojèsoume ìti α 6= ±1. Ja de�xoume ìti

λ =

{

κ an 1 ≤ n ≤ µ

pn−µκ an n ≥ µ.'Estw n ≤ µ. 'Opw prin èqoume ακ ≡ 1mod pµ kai �ra ακ ≡ 1mod pn.Sunep¸ λ|κ. All� αλ ≡ 1mod pn, opìte αλ ≡ 1mod p kai �ra κ|λ apì toopo�o prokÔptei ìti κ = λ. 'Estw n ≥ µ. 'Eqoume ακ = 1 + pµt me p ∤ t, µ ≥ 1.An n ≥ µ tìte isqÔei(*) (ακ)pn−µ

= 1 + tnpn, me p ∤ tn.

3.5. Prwtarqikè R�ze 159Efarmìzoume epagwg sto n. Gia n = µ isqÔei. 'Estw n ≥ µ, tìte(ακ)p

n+1−µ= (1 + tnp

n)p

= 1 +

(

p

1

)

tnpn +

(

p

2

)

t2np2n +

p∑

i=3

(

p

i

)

tinpin

= 1 + tnpn+1 +

p− 1

2t2np

2n+1 +

p∑

i=3

(

p

i

)

tinpin.Kaj¸ n ≥ 1, èqoume 2n + 1 ≥ n + 2 kai ep�sh in ≥ 3n ≥ n + 2 afoÔ i ≥ 3.Sunep¸ èqoume

(ακ)pn+1−µ

= 1 + tnpn+1 + unp

n+2 = 1 + tn+1pn+1ìpou tn+1 = tn + unp, me p ∤ tn+1.Dhlad isqÔei h isotim�a αpn−µ ≡ 1mod pn. Sunep¸ λ|pn−µκ. All� epeid

αλ ≡ 1mod pn e�nai kai αλ ≡ 1mod p, opìte κ|λ, dhlad λ = κλ1. 'Araκλ1|pn−µκ pou shma�nei ìti λ1|pn−µ. Opìte λ1 = ps, 0 ≤ s ≤ n−µ. De�qnoumeìti s = n− µ. Prohgoumènw h sqèsh (∗) apede�qjei gia k�je n, n ≥ µ, opìteisqÔei kai gia r = µ+ s ≥ µ, dhlad èqoume

ακpr−µ= ακps

= 1 + trpµ+s, p ∤ tr.'Eqoume ìmw αλ = ακλ1 = ακps ≡ 1mod pn. Sunep¸ pn|trpµ+s kai �ra

pn|pµ+s, opìte n ≤ µ+ s s ≥ n− µ. Telik� s = n− µ.Parade�gmata.1. 'Eqoume 7 ≡ −1mod 4, �ra o4(7) = 2. E�nai de 72 − 1 = 48 = 24 · 3 kaisunep¸ µ = 4. 'Arao2n(7) =

{

2 an 2 ≤ n ≤ 4

2n−4 · 2 = 2n−3 an n ≥ 4.2. 'Eqoume 5 ≡ 1mod 4, �ra o4(5) = 1. E�nai de 5 − 1 = 22 kai sunep¸ µ = 2. Opìte

o2n(5) =

{

1 an 2 ≤ n ≤ 2, dhlad an n = 2

2n−2 × 1 = 2n−2 an n ≥ 2.

160 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze 3. O p = 433 e�nai pr¸to kai op(137) = 18. Ep�sh èqoume 13718 6≡1mod 4332 opìte µ = 1.'Ara o433n(137) = 433n−1 · 18 an n ≥ 1.4. Upolog�zoume thn t�xh on(89) ìpou n = 24 · 32 · 53:

on(89) = [o24(89), o32(89), 053 (89)].E�nai o24(89) = 2, o32(89) = 2, o53(89) = 50,opìte on(89) = [2, 2, 50] = 50.W mia efarmog tou Jewr mato 3.5.1 apodeiknÔoume to ex 3.5.2 Je¸rhma. 'Estw p èna perittì pr¸to arijmì . An o p diaire� tonα2n

+ 1, α ∈ Z, tìte p ≡ 1mod 2n+1 (gia par�deigma an p|α2 + 1 tìte p ≡1mod 4).Apìdeixh. 'Estw α2n ≡ −1mod p, opìte α2n+1 ≡ 1mod p. Sunep¸ h t�xhκ = op(α) diaire� ton 2n+1 kai �ra κ = 2t, 0 ≤ t ≤ n + 1. De�qnoume ìtit = n + 1. Upojètoume ìti den isqÔei, dhlad t ≤ n. All� tìte ja e�qameα2t ≡ 1mod p kai �ra (α2t

)2n−t ≡ 1mod p α2n ≡ 1mod p. All� d�detai ìti

α2n ≡ −1mod p kai sunep¸ prèpei t = n + 1 kai κ = 2n+1. Apì to 3.5.1 ii)prokÔptei telik� ìti 2n+1|p− 1.3.5.3 Pìrisma. Up�rqoun �peiroi to pl jo pr¸toi th morf 2n+1κ+ 1.Apìdeixh. Upojètoume ìti p1, p2, . . . , ps e�nai pr¸toi th morf 2n+1κ + 1.'Estw

N = (2p1 · · · ps)2n

+ 1.Tìte an p e�nai èna pr¸to diairèth tou N , o p e�nai perittì kai apì to3.5.2 autì èqei th morf 2n+1κ + 1. All� p 6= pi, i = 1, 2, . . . , s afoÔ p|N .'Ara èqoume brei ènan pr¸to th morf 2n+1κ + 1 pou e�nai di�foro twn pi,i = 1, . . . , s.

3.5. Prwtarqikè R�ze 161T¸ra apì to Je¸rhma tou Euler gnwr�zoume ìti h mègisth dunat t�xhpou mpore� na èqei èna akèraio αmodm e�nai ϕ(m). E�dame ìti gia m = 8kanèna akèraio den èqei t�xh ϕ(8) = 4. Gia m = 12, h t�xh twn antistrèyimwnkl�sewn 5, 7 kai 11 e�nai 2. Dhlad kai s� aut thn per�ptwsh kam�a kl�sh denèqei t�xh �sh me ϕ(12) = 4. Gia m = 4 m = 6, up�rqei mìno mia antistrèyimhkl�sh, di�forh th 1modm, 3mod 4 kai h 5mod 6 kai sunep¸ anagkastik�èqei t�xh �sh me ϕ(4) = ϕ(6) = 2. An m = 9, oi antistrèyime 6= 1 kl�sei e�naioi 2, 4, 5, 7 kai 8mod 9 pou oi t�xei tou e�nai ant�stoiqa �se me 6, 3, 6, 2.'Opw ep�sh an m = 10, oi antistrèyime 6= 1 kl�sei e�nai 3, 7 kai 9 pou èqount�xh 4, 4 kai 2 ant�stoiqa. Blèpoume ìti sti dÔo autè peript¸sei up�rqounkl�sei pou èqoun t�xh �sh me ϕ(9) = 6 kai ϕ(10) = 4.'Otan up�rqei èna akèraio α pou h t�xh tou modm isoÔtai me ϕ(m) tìtelème ìti o α e�nai mia prwtarqik r�za modulo m. Oi prwtarqikè r�ze pa�zounshmantikì rìlo sth Jewr�a Arijm¸n (ìpw ja g�nei fanerì pio k�tw) all� kaigenik¸tera se ìla ta majhmatik�. Kur�w ìmw èqoun efarmogè sta legìmena“Diakrit� Majhmatik�”.Tr�a e�nai ta basik� all� kai eÔloga erwt mata pou ege�rontai.1o Gia poiou fusikoÔ arijmoÔ m up�rqoun prwtarqikè r�ze modm;2o 'Otan up�rqoun prwtarqikè r�ze , pìse e�nai autè ;3o 'Otan up�rqoun prwtarqikè r�ze poie e�nai autè ;H ap�nthsh tou 2oυ erwt mato e�nai apl kai bas�zetai ston orismì th prwtarqik r�za kai sto 3.5.1. Pr�gmati, èna isodÔnamo orismì gia ti prwtarqikè r�ze e�nai o ex . 'Ena akèraio α e�nai prwtarqik r�za an kaimìnon an oi ϕ(m) akèraioi 1, α, α2, . . . , αϕ(m)−1 apoteloÔn èna pl re sÔsth-ma antipros¸pwn pr¸twn kl�sewn upolo�pwn modm. AfoÔ an α e�nai miaprwtarqik r�za tìte sÔmfwna me to 3.5.1 iii) auto� oi ϕ(m) akèraioi e�nai an�dÔo mh isìtimoi. Ant�strofa, an oi ϕ(m) auto� arijmo� e�nai mh isìtimoi tìteακ 6≡ 1modm, 1 ≤ κ ≤ ϕ(m) − 1 kai epeid αϕ(m) ≡ 1modm, o α èqei t�xhϕ(m). T¸ra upojètoume ìti up�rqei mia prwtarqik r�za αmodm. Apì to

162 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze 3.5.1 h t�xh tou ακ modulo m isoÔtai me ϕ(m)/(κ, ϕ(m)). Sunep¸ o ακ e�naimia prwtarqik r�za an kai mìnon an (κ, ϕ(m)) = 1, afoÔ an o ακ e�nai prwtar-qik r�za tìte ϕ(m)/(κ, ϕ(m)) = ϕ(m) kai �ra (κ, ϕ(m)) = 1. Ant�strofa an(κ, ϕ(m)) = 1, kai λ e�nai h t�xh tou ακ, prèpei ϕ(m)|κλ, apì to 3.5.1 ii), opìteϕ(m)|λ kai sunep¸ ϕ(m) = λ, afoÔ to λ w t�xh mia kl�sh modm prèpeina diaire� to ϕ(m).T¸ra epeid oi akèraioi 1, α, α2, . . . , αϕ(m)−1 apoteloÔn èna pl re sÔsthmaantipros¸pwn pr¸twn kl�sewn upolo�pwn modm, an β e�nai mia prwtarqik r�za modm aut prèpei na e�nai isìtimh me mia dÔnamh ακ, 0 ≤ κ ≤ ϕ(m) − 1,pou èqei t�xh ϕ(m). Dhlad o akèraio β e�nai prwtarqik r�za modm an kaimìnon an β ≡ ακ modm, 0 ≤ κ ≤ ϕ(m) − 1, (κ, ϕ(m)) = 1. To pl jo aut¸ntwn dun�mewn tou α e�nai �so me ϕ(ϕ(m)). 'Etsi èqoume apode�xei to ex .3.5.4 Je¸rhma. An up�rqei mia prwtarqik r�za modm tìte up�rqoun akri-b¸ ϕ(ϕ(m)) prwtarqikè r�ze modm.Par�deigma. 'Estw m = 10. To 3 e�nai prwtarqik r�za mod10. Sune-p¸ ìle oi antistrèyime kl�sei e�nai oi 1mod 10, 3mod 10, 32 mod 10 kai33 mod10. Ap� autè ti ϕ(10) = 4 kl�sei oi ϕ(ϕ(10)) = 2 e�nai prwtar-qikè r�ze kai autè e�nai oi 3mod 10 kai 33 mod 10, afoÔ (2, 4) = 2 kai(3, 4) = 1. An m = 11, to 2 e�nai prwtarqik r�za mod 11. Apì ti dun�-mei 1, 2, 22, 23, 24, 25, 26, 27, 28, 29, oi dun�mei 2, 23, 27 kai 29 e�nai akrib¸ oiprwtarqikè r�ze mod11 isodÔnama oi akèraioi 2, 8, 7, 6 e�nai oi prwtarqikè r�ze mod11 afoÔ 27 ≡ 7mod 11 kai 29 ≡ 6mod 11.Sqetik� me to 1o er¸thma ja de�xoume ìti up�rqoun prwtarqikè r�ze modm an kai mìnon an o m e�nai 2, 4, pn 2pn, ìpou p e�nai pr¸to arij-mì . Dustuq¸ den up�rqei apìdeixh autoÔ tou apotelèsmato pou na e�naikataskeuastik . Sunep¸ den ma d�nei kam�a plhrofor�a poie akrib¸ e�nai oiprwtarqikè r�ze kai ètsi to 3o er¸thma paramènei èw s mera anap�nthto. Gi�autì to er¸thma up�rqei h ex sqetik eikas�a tou Emil Artin. An α 6= ±1kai o α den e�nai èna tèleio tetr�gwno, tìte up�rqoun �peiroi to pl jo pr¸toi

3.5. Prwtarqikè R�ze 163arijmo� p tètoioi ¸ste o α e�nai mia prwtarqik r�za modulo p. Aut h eikas�ae�nai mia gen�keush th eikas�a tou Gauss (up�rqei sto sÔggramma tou Disqui-

sitiones Arithmeticae) pou anafèrei ìti up�rqoun �peiroi to pl jo pr¸toi pgia tou opo�ou to 10 e�nai prwtarqik r�za mod p. H eikas�a tou Artin pa-ramènei èw s mera anap�nthth kai sundèetai me thn “Upìjesh tou Riemann”.Shmei¸noume ìti o periorismì pou t�jetai sto α sthn eikas�a aut e�nai anag-ka�o diìti profan¸ to ±1 den e�nai prwtarqik r�za gia �peirou pr¸tou ,afoÔ h t�xh tou 1 e�nai 1 kai tou −1 e�nai 2 (gia p 6= 2). An e�nai de α = β2,gia k�poio β ∈ Z, tìteα

p−12 =

(

βp−12

)2= βp−1 ≡ 1mod pkai sunep¸ o α den mpore� na e�nai mia prwtarqik r�za mod p (an p|β, tìte

p|α kai fusik� αp−1 6≡ 1mod p).T¸ra apodeiknÔoume, se diaforetik� b mata, to sqetikì me to 1o er¸thmaapotèlesma pou anafèrjhke prin l�go.3.5.5 Je¸rhma. 'Estw m = 2n. Tìte den up�rqoun prwtarqikè r�ze modm, ektì an m = 2 4.Apìdeixh. Epeid ϕ(2) = 1, ϕ(4) = 2 kai 11 ≡ 1mod 2, 32 ≡ 1mod 4, up�rqounprwtarqikè r�ze modm gia m = 2 kai m = 4.'Estw m = 2n, n ≥ 3. Gia na e�nai mia kl�sh αmod2n antistrèyimh prèpeikai arke� o α na e�nai perittì . An αmod2n e�nai mia tètoia kl�sh ja de�xoumeìti isqÔei

α2n−2 ≡ 1mod 2n,opìte o α den mpore� na e�nai prwtarqik r�za mod 2n, afoÔ to pl jo twnsqetik� pr¸twn pro ton 2n arijm¸n r, 1 ≤ r ≤ 2n, e�nai ϕ(2n) = 2n−1.Dhlad den up�rqoun prwtarqikè r�ze mod2n. Efarmìzoume epagwg ston. Gia n = 3, upenjum�zoume apì prohgoÔmeno par�deigma ìti

12 ≡ 32 ≡ 52 ≡ 72 ≡ 1mod 8 kai ϕ(8) = 4.

164 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze Sunep¸ gia n = 3 isqÔei. Upojètoume ìtiα2n−2 ≡ 1mod 2n.Autì shma�nei ìti α2n−2

= 1 + 2nλ, gia λ ∈ Z. Opìte èqoumeα2n−1

= (1 + 2nλ)2 = 1 + 2n+1λ+ 22nλ2 = 1 + 2n+1(λ+ 2n−1λ2) ≡ 1mod 2n+1.ApodeiknÔoume t¸ra thn Ôparxh prwtarqik¸n riz¸n mod p, p pr¸to . Jad¸soume dÔo apode�xei . H pr¸th e�nai apl kai �mesh en¸ h deÔterh, an kaiden e�nai tìso apl , e�nai ousiastik kaj¸ anafèretai sthn t�xh ìlwn twnantistrèyimwn kl�sewn mod p.3.5.6 Je¸rhma. An m = p e�nai èna pr¸to arijmì tìte up�rqoun ϕ(p−1)prwtarqikè r�ze mod p.1η Apìdeixh. 'Estw ℓ h mègisth dunat t�xh metaxÔ ìlwn twn t�xewn twn kl�-sewn mod p. An α e�nai èna akèraio me p ∤ α, tìte h t�xh o(α) diaire� ton ℓ,apì to 3.5.1. Sunep¸ αℓ ≡ 1mod p. 'Ara ìle oi antistrèyime kl�sei mod p,pou e�nai se pl jo ϕ(p) = p− 1, e�nai lÔsei th poluwnumik isotim�a

xℓ − 1 ≡ 0mod ph opo�a èqei to polÔ ℓ lÔsei sÔmfwna me to Je¸rhma Lagrnage 3.3.3. 'Araϕ(p) ≤ ℓ. All� ℓ|ϕ(p) kai sunep¸ ℓ = ϕ(p) ( �mesa, sÔmfwna me to 3.3.6,afoÔ ℓ|ϕ(p) = p− 1, xℓ − 1 ≡ 0mod p èqei akrib¸ ℓ lÔsei , �ra ℓ = ϕ(p) =

p− 1).Gia th 2η apìdeixh qreiazìmaste èna apotèlesma th Jewr�a Arijm¸n gnw-stì w TÔpo tou Gauss.3.5.7 Je¸rhma (Gauss). 'Estw n ∈ N kai 1 = d0, d1, . . . , dk = n ìloi oidiairète tou n. Tìte isqÔeiϕ(d0) + ϕ(d1) + · · · + ϕ(dk) = n,

(en suntom�a ∑

d|n

ϕ(d) = n

)

.

3.5. Prwtarqikè R�ze 165Apìdeixh. Up�rqoun di�fore apode�xei autoÔ tou apotelèsmato . M�a apìautè e�nai h ex . Diamer�zoume to sÔnolo An = {1, 2, . . . , n} w pro thsqèsh isodunam�a : x, y ∈ An e�nai isodÔnama an (x, n) = (y, n). Sunep¸ anx ∈ An kai o mègisto koinì diairèth (x, n) = d, h kl�sh isodunam�a sthnopo�a an kei o x e�nai

Ad,n = {y ∈ Sn/(y, n) = d}kai èqoumeAn =

˙⋃

d|n

Ad,n.Sunep¸ n = |An| =∑

d|n

|Ad,n|.Isqurizìmaste ìti |Ad,n| = ϕ(n

d

). Pr�gmati, èstw y ∈ Ad,n, kai y = td.Opìte (

t,n

d

)

= 1, dhlad t ∈ A1, nd. JewroÔme thn apeikìnish

Ad,n → A1, nd, y → t.Aut e�nai 1−1 kai ep�, afoÔ gia y1, y2 ∈ Ad,n, y1 = t1d, y2 = t2d e�nai y1 6= y2 ankai mìnon an t1 6= t2 kai gia t ∈ A1, n

de�nai (td, n) = d. H teleuta�a isìthta isqÔeidiìti apì thn (

t,n

d

)

= 1, lìgw th 1.1.12 iv), èqoume d(

t,n

d

)

= (td, n) = d.Dhlad gia t ∈ A1, ndup�rqei to y = td ∈ Ad,n pou antistoiqe� sto t. Sunep¸ èqoume

n =∑

d|n

ϕ(n

d

)

.All� kaj¸ to d diatrèqei ìlou tou diairète d0, d1, . . . , dk tou n, to n

ddiatrèqei ep�sh ìlou tou diairète dk, dk−1, . . . , d0 tou n (jètonta ndi

= dκ−i,i = 1, . . . , κ). 'Ara telik�

n =∑

d|n

ϕ(n

d

)

=∑

d|n

ϕ(n).

166 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze Shmei¸noume ìti h teleuta�a parat rhsh sthn apìdeixh ìti o d kai o n

dam-fìteroi diatrèqoun ìlou tou jetikoÔ diairète tou n e�nai èna sunhjismèno trìpo (duðkì ) skèyh sth jewr�a Arijm¸n.

2η Apìdeixh tou 3.5.6. Gia k�je jetikì diairèth d tou p−1, jewroÔme to sÔnoloSd = {α/1 ≤ α ≤ p− 1, o(α) = d}.Profan¸ èqoume thn ex diamèrish (xènh ènwsh) tou sunìlouAp−1 = {1, 2, . . . , p−

1},Ap−1 =

˙⋃

d|n

Sd.Skopì ma e�nai na deiqje� ìti Sp−1 6= ∅.T¸ra de�qnoume ìti |Sd| = ϕ(d) ìtan Sd 6= ∅. Pr�gmati èstw Sd 6= ∅ kaiα ∈ Sd, tìte oi d to pl jo akèraioi 1, α, α2, . . . , αd−1 e�nai an� dÔo mh isìti-moi mod p kai e�nai lÔsei mod p th poluwnumik isotim�a xd − 1 ≡ 0mod ph opo�a apì to Je¸rhma tou Lagrange 3.3.3 èqei to polÔ d mh-isìtime lÔsei mod p. 'Ara oi d kl�sei 1mod p, αmod p, . . . , αd−1 mod p e�nai akrib¸ ìle oilÔsei aut ( �mesa, epeid d|p−1, apì to 3.3.6, h xd−1 ≡ 0mod p èqei akri-b¸ d mh-isìtime lÔsei mod p pou e�nai oi 1mod p, αmod p, . . . , αd−1 mod p).All� k�je stoiqe�o tou Sd e�nai mia apì ti lÔsei mod p th xd − 1 ≡ 0mod p.Sunep¸ k�je β ∈ Sd e�nai isìtimo me k�poia dÔnamh ακ tou α me 1 ≤ κ ≤ d−1.Epeid o β èqei t�xh d kai o ακ èqei t�xh d pou autì shma�nei, lìgw tou 3.5.1 iv),ìti (κ, d) = 1. 'Ara

Sd = {β ∈ Ap−1/β ≡ ακ mod p, 1 ≤ κ ≤ d− 1, (κ, d) = 1}.Sunep¸ |Sd| = ϕ(d), an Sd 6= ∅. All� epeid ta sÔnola Sd diamer�zoun tosÔnolo Ap−1, èqoume p − 1 = |Ap−1| =∑

d|p−1

|Sd|. Ep�sh apì to 3.5.7 èqoumep− 1 =

∑

d|p−1

ϕ(d). 'Ara∑

d|p−1

(ϕ(d) − |Sd|) =∑

d|p−1

ϕ(d) −∑

d|p−1

|Sd| = (p − 1) − (p − 1) = 0.

3.5. Prwtarqikè R�ze 167Epeid ϕ(d)−|Sd| ≥ 0, gia k�je d|p−1, prokÔptei ìti ϕ(d)−|Sd| = 0, gia k�jed|p − 1. 'Ara |Sd| = ϕ(d) gia k�je d|p− 1. Idiaitèrw gia d = p− 1, pa�rnoumeϕ(p − 1) = |Sp−1|, dhlad up�rqoun ϕ(p − 1) prwtarqikè r�ze mod p.Parat rhsh. Apì th 2η apìdeixh prokÔptei to isqurìtero apotèlesma apìeke�no pou diatup¸netai sto Je¸rhma 3.5.6: An d e�nai èna diairèth tou p −1 tìte up�rqoun akrib¸ ϕ(d) mh-isìtimoi mod p akèraioi pou èqoun t�xh d.Fereipe�n, an 4|p−1, tìte up�rqoun akrib¸ dÔo mh-isìtimoi akèraioi pou èqount�xh 4. An α e�nai èna ap� autoÔ tìte o �llo e�nai o −α. 'Eqoume deα4 ≡ 1mod p isodÔnama (α2 − 1)(α2 + 1) ≡ 0mod p pou shma�nei ìti α2 ≡1mod p α2 ≡ −1mod p. Epeid h t�xh tou α e�nai 4 den mpore� na isqÔei hα2 ≡ 1mod p. Sunep¸ prèpei α2 ≡ −1mod p. Autì ma d�nei mia �llh apìdeixhth mia kateÔjunsh tou Jewr mato 2.3.12, dhlad ìti an p = 4n + 1 tìte hx2 ≡ −1mod p èqei lÔsh.Ax�zei na parathrhje� ep�sh ìti an kai anamènetai ìti o arijmì |Sd| na exar-t�tai apì to d kai apì to p blèpoume ìti exart�tai mìno apì to d (upojètonta ìti d|p− 1, dhlad e�nai anex�rthto tou p. Gia par�deigma, an d = 6 up�rqounakrib¸ |S6| = 2 mh isìtimoi akèraioi pou èqoun t�xh 6 gia ìlou tou pr¸tou p gia tou opo�ou 6|p− 1 p.q. p = 7, 13, 19, 31, . . . .Parade�gmata.1. 'Estw p = 17. E�nai ϕ(17) = 16 kai �ra up�rqoun ϕ(16) = 8 prwtarqikè r�ze mod17. Oi diairète tou 16 e�nai d = 1, 2, 4, 8, 16. 'Ara up�rqoun

ϕ(1) = 1, ϕ(2) = 1, ϕ(4) = 2, ϕ(8) = 4 kai ϕ(16) = 8 mh-isìtimoi mod17akèraioi pou èqoun t�xh 1, 2, 4, 8 kai 16mod 17 ant�stoiqa. Met� apìmerikoÔ upologismoÔ br�skoume ìti oi dun�mei 3κ tou 3 ma d�noun ènapl re sÔsthma antipros¸pwn pr¸twn kl�sewn upolo�pwn mod 17:1, 3, 32 ≡ 9mod 17, 33 ≡ 10mod 17, 34 ≡ 13mod 17, 35 ≡ 14mod 17,36 ≡ 15mod 17, 37 ≡ 11mod 17, 38 ≡ 16mod 17, 39 ≡ 14mod 17,310 ≡ 8mod 17, 311 ≡ 7mod 17, 312 ≡ 4mod 17, 313 ≡ 12mod 17,314 ≡ 2mod 17, 315 ≡ 6mod 17.

168 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze Epeid gia k�je κ = 0, 1, . . . , 15 oi arijmo� 16

(κ, 16)e�nai oi t�xei twndun�mewn 3κ, br�skoume ìti

S1 ={1}S2 ={16}S4 ={4, 13}S8 ={2, 8, 9, 15}S16 ={3, 5, 6, 7, 10, 11, 12, 14}.2. Me autì to par�deigma upodeiknÔoume thn qrhsimìthta twn prwtarqik¸nriz¸n d�nonta mia �llh apìdeixh tou Jewr mato tou Wilson.'Estw p èna perittì pr¸to arijmì kai α mia prwtarqik r�za. Tìte oiarijmo� 1, 2, . . . , p− 1 e�nai isìtimoi mod p me tou α,α2, . . . , αp−1. 'Ara

(p− 1)! ≡ αα2 · · ·αp−1 mod p ≡ α1+2···+p−1 ≡ α(p−1)p

2 ≡(

αp−12

)pmod p.Apì to Je¸rhma tou Fermat o α p−1

2 e�nai lÔsh th x2 ≡ 1mod p kai oilÔsei aut e�nai oi 1 kai −1mod p. An α p−12 ≡ 1mod p, tìte o α den ja tan prwtarqik r�za mod p, sunep¸

αp−12 ≡ −1mod p.'Etsi prokÔptei to Je¸rhma tou Wilson:

(p− 1)! ≡ (−1)p ≡ −1mod p.Ep�sh , tautìqrona èqoume apode�xei, ìti opoted pote o α e�nai mia prw-tarqik r�za mod p tìte isqÔei:α

p−12 ≡ −1mod p.3.5.8 Je¸rhma. 'Estw p èna perittì pr¸to arijmì kai α mia prwtarqik r�za mod p. Tìte

3.5. Prwtarqikè R�ze 169i) an

αp−1 6≡ 1mod p2tìte o α e�nai mia prwtarqik r�za mod pn, gia k�je n ≥ 1.ii) an

αp−1 ≡ 1mod p2,tìte (α + p)p−1 6≡ 1mod p2 kai �ra o α + p e�nai mia prwtarqik r�zamod pn, gia n ≥ 1.Sunep¸ gia k�je perittì pr¸to arijmì p p�nta up�rqoun prwtarqikè r�ze

mod pn, gia k�je n ≥ 1.Apìdeixh. i) 'Eqoume apì thn upìjesh ìti p2 ∤ αp−1 − 1. SÔmfwna me thnParat rhsh 5 (sel. 123) ja èqoume opn(α) = pn−1op(α) = pn−1(p−1) = ϕ(pn).ii) IsqÔei

(α+ p)p−1 = αp−1 +

(

p− 1

1

)

αp−2p+ p2κ

≡(

αp−1 + (p− 1)αp−2p)

mod p2

≡ (1 − αp−2p)mod p2

6≡ 1mod p2(afoÔ diaforetik� o α den ja tan prwtarqik r�za mod p). Ep�sh epeid o αe�nai mia prwtarqik r�za mod p kai o α + p e�nai mia prwtarqik r�za mod p,afoÔ (α+ p)s ≡ αs mod p, gia k�je s ≥ 1. Sunep¸ sÔmfwna me to i), o α+ pe�nai mia prwtarqik r�za mod pn, n ≥ 1.Par�deigma. p = 71. Br�skoume eÔkola ìti o71(11) = 70. Ep�sh 712|1170−1, opìte 712 ∤ (11 + 71)70 − 1 kai �ra to 82 e�nai mia prwtarqik r�za mod71n,gia ìla ta n ≥ 1.

170 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze 3.5.9 Je¸rhma. 'Estw p èna perittì pr¸to arijmì kai α mia prwtarqik r�za mod pn. Tìte o akèraio β =

α an o α e�nai perittì α+ pn an o α e�nai �rtio .e�nai mia prwtarqik r�za mod 2pn, gia ìla ta n.Apìdeixh. Kat� arq� èqoume (β, 2pn) = 1 kai ep�sh opn(β) = opn(α) =

ϕ(pn). 'Ara, sÔmfwna me thn Parat rhsh 5 (sel. 122), èqoume o2pn(β) =

[o2(β), opn(β)] = [1, ϕ(pn)] = ϕ(pn) = ϕ(2pn).Par�deigma. o3(2) = 2, 3|22 − 1, 32 ∤ 22 − 1. 'Ara o3n(2) = ϕ(3n), dhlad o 2 e�nai mia prwtarqik r�za mod3n. Sunep¸ o 2 + 3n e�nai mia prwtarqik r�za mod2 · 3n.'Ew t¸ra èqoume de�xei ìti an m = 2, 4, pn, 2pn, ìpou p e�nai èna perittì pr¸to , tìte up�rqoun (ϕ(ϕ(m)) to pl jo ) prwtarqikè r�ze modm. T¸raja de�xoume ìti isqÔei kai to ant�strofo.3.5.10 L mma. 'Estw m = m1m2 ìpou (m1,m2) = 1 kai m1 > 2, m2 > 2.Tìte den up�rqei kam�a prwtarqik r�za modm.Apìdeixh. 'Estw α èna akèraio sqetik� pr¸to pro ton m, dhlad h kl�shαmodm e�nai antistrèyimh. Tìte om(α) = [om1(α), om2(α)]. Gnwr�zoume ìtiom1(α)|ϕ(m1) kai om2(α)|ϕ(m2) kai �ra [om1(α), om2(α)]|[ϕ(m1), ϕ(m2)].All� oi ϕ(m1) kai ϕ(m2) e�nai �rtioi arijmo�, sunep¸ isqÔei

[ϕ(m1), ϕ(m2)] < ϕ(m1)ϕ(m2).Telik� èqoumeom(α) < ϕ(m1)ϕ(m2) = ϕ(m1m2),lìgw tou 2.4.6.W pìrisma autoÔ èqoume to ex .

3.6. De�kte kai Diwnumikè Isotim�e 1713.5.11 Je¸rhma. 'Estwm > 1. Tìte an up�rqei mia prwtarqik r�za modmprèpei o m na e�nai th morf 2, 4, pn 2pn, ìpou p e�nai perittì pr¸to arijmì .Apìdeixh. An o m e�nai �rtio tìte apì to l mma prèpei m = 2t 2pt. Anm = 2t apì to 3.5.5 prèpei t = 1 2. An m e�nai perittì tìte apì to l mmaprèpei m = pt.Shmei¸noume ìti to Je¸rhma 3.5.11 apede�qjei to 1801 apì ton Gauss en¸to 3.5.6 apede�qjei apì ton Euler kai ton Legendre.3.6 De�kte kai Diwnumikè Isotim�e Oi prwtarqikè r�ze modm (ìtan up�rqoun) kai h idiìthta 3.5.1 i) ma epitrè-poun na metatrèyoume ta pollaplasiastik� probl mata modm se prosjetik�probl mata modϕ(m). Autì epitugq�netai me thn ènnoia tou de�kth enì akè-raiou arijmoÔ w pro mia prwtarqik r�za modm pou ìrise o Gauss w ex :'Estw g mia prwtarqik r�za modm. Tìte oi akèraioi 1, g, g2, . . . , gϕ(m)−1 apo-teloÔn èna pl re sÔsthma antipros¸pwn pr¸twn kl�sewn upolo�pwn modm.Sunep¸ an α e�nai èna akèraio sqetik� pr¸to pro ton m, up�rqei èna monadikì akèraio κ, 0 ≤ κ ≤ ϕ(m)−1, gia ton opo�o isqÔei α ≡ gκ modm. Oakèraio κ onom�zetai o de�kth tou α w pro th b�sh gmodm kai gr�foumeκ ≡ ind gα apl� κ = indα ìtan gnwr�zoume poia e�nai h b�sh. Apì tonorismì kai to 3.5.1 i) prokÔptei ìti gia dÔo akèraiou α kai β (pou p�nta tou jewroÔme sqetik� pr¸tou pro ton m) isqÔei

indα = indβ an kai mìnon an α ≡ βmodm.Dhlad ìloi oi akèraioi se mia dosmènh kl�sh modm èqoun ton �dio de�kth.'Etsi, ìtan jewroÔme p�nake stou opo�ou anagr�fontai oi timè gia tou de�kte indα, arke� na jewroÔme mìno eke�nou tou akèraiou α tou sqetik�pr¸tou pro ton m kai 1 ≤ α ≤ m − 1. Gia par�deigma, an m = 13 eÔkola

172 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze br�skoume ìti o 2 e�nai prwtarqik r�za kai èqoume ton p�nakaOi timè tou ind2α (mod 13)

α: 1 2 3 4 5 6 7 8 9 10 11 12ind2α: 0 1 4 2 9 5 11 3 8 10 7 6Autì o p�naka de�qnei ìti o 6, o 7 kai o 11 e�nai ep�sh prwtarqikè r�ze

mod 13 (afoÔ autè e�nai isìtime me tou 25, 211 kai 27 ant�stoiqa). 'Etsi,ep�sh èqoume to Oi timè tou ind6α (mod 13)

α: 1 2 3 4 5 6 7 8 9 10 11 12ind6α: 0 5 8 10 9 1 7 3 4 2 11 6To epìmeno je¸rhma perilamb�nei ti basikè idiìthte th sun�rthsh ind .3.6.1 Je¸rhma.

i) indgαβ ≡ (indgα+ indgβ)modϕ(m)

ii) indgαn ≡ nindgαmodϕ(m), n ≥ 1

iii) indgα ≡ (indg1indgg1)modϕ(m)

iv) indgg1indg1g2 · indgg1 ≡ 1modϕ(m)

v) ind1 = 0

vi) indgg = 1

vii) indg(−1) =1

2ϕ(m), m > 2.Apìdeixh. i) 'Estw indgα = κ kai indgβ = λ, dhlad α ≡ gκ modm kai

β ≡ gλ modm. Opìte αβ ≡ gκ+λ modm. 'Estw indαβ = µ, dhlad αβ ≡gµ modm. 'Ara gµ ≡ gκ+λ modm. Opìte apì to 3.5.1 i) prokÔptei ìti µ ≡κ+ λmodϕ(m).

3.6. De�kte kai Diwnumikè Isotim�e 173H ii) prokÔptei apì thn i). Gia thn iii) jètoume indg1α = κ kai indgg1 = λ,opìte α = gκ1 modm kai g1 ≡ gλ modm. 'Etsi èqoume α ≡ gκλ modm ≡

gindgα modm. 'Ara κλ ≡ (indgα)modϕ(m).Me ton �dio trìpo prokÔptei kai h iv). Oi v) kai vi) e�nai profane� .vii) An m = 4, tìte 1

2ϕ(m) = 1, g = 3 kai −1 ≡ g1 mod4. Opìte isqÔeigia m = 4. Se opoiad pote �llh per�ptwsh prèpei m = pn 2pn ìpou p e�naiperittì pr¸to arijmì kai n ≥ 1. Se k�je per�ptwsh èqoume(*) gϕ(m) − 1 =(

g12ϕ(m) − 1

) (

g12ϕ(m) + 1

)

≡ 0modm.De�qnoume ìti ((

g12ϕ(m) − 1

)

, pn)

= 1. Pr�gmati, an den �sque autì, tìtep|g 1

2ϕ(m)−1. All� tìte p ∤ g

12ϕ(m) +1, afoÔ (

g12ϕ(m) + 1

)

−(

g12ϕ(m) − 1

)

= 2kai p ≥ 3. Opìte apì thn prohgoÔmenh sqèsh (∗) ja proèkupte ìti pn|g 12ϕ(m)−1kai epeid o g e�nai perittì an m = 2pn, ja e�qame

g12ϕ(m) − 1 ≡ 0modm.Dhlad o g den ja tan mia prwtarqik r�za ìpw èqoume upojèsei apì thnarq . Sunep¸ apì thn sqèsh (∗) prokÔpteig

12ϕ(m) ≡ −1modm.Parathr sei .1. Apì to Je¸rhma 3.6.1 prokÔptei ìti ind(n − α) ≡ ind(−α) ≡ ind(−1) +

indα

(

1

2ϕ(m) + indα

)

modϕ(m). Opìte stou p�nake twn deikt¸n apìti ϕ(m) timè oi misè mporoÔn na paralhfjoÔn.2. To prohgoÔmeno Je¸rhma 3.6.1 upodhl¸nei ìti h sun�rthsh ind èqei ti �die idiìthte me th logarijmik sun�rthsh log. Gi� autì to lìgo o de�-kth indα merikè forè onom�zetai “diakritì log�rijmo ” tou α. A shmeiwje� ìti ìpw o log(0) den up�rqei, to �dio kai o indgα den or�zetaian α ≡ 0modm.

174 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze Efarmogè .1. Upologismì dun�mewn. Na upologisje� o mikrìtero jetikì akèraio pou e�nai isìtimo me ton 5106modulo 13, dhlad na breje� o r, 1 ≤ r ≤ 12,me r ≡ 5106

mod13. JewroÔme thn prwtarqik r�za 2mod 13 kai èqoume2ind25106 ≡ 5106 ≡ r ≡ 2ind2r mod 13.'Ara ja prèpei ind25

106 ≡ ind2rmod12. Opìte, apì to 3.6.1 ii), 106ind25 ≡(−2)6 · 9 ≡ 0mod 12. Ap� ton prohgoÔmeno p�naka, ind21 = 0 ( apìto 3.6.1 v)). Opìte r = 1. Sunep¸ 5106 ≡ 1mod 13. To �dio ja e�-qame an jewroÔsame w prwtarqik r�za thn 6 ( opoiad pote �llh):6ind65106 ≡ 5106 ≡ r ≡ 6ind6r mod 13. 'Ara ind65

106 ≡ ind6rmod12.Opìte 106ind65 ≡ (−2)6 · 9 ≡ 0mod 12.2. Ep�lush th αx ≡ βmodm, ìtan up�rqoun prwtarqikè r�ze modm.An m|β, tìte oi lÔsei e�nai oi κm, κ ∈ Z. 'Estw d = (α,m), tìtegnwr�zoume ìti h isotim�a èqei lÔsh an kai mìnon an d|β. Upojètoume ìtiβ 6≡ 0modm kai ìti d|β. JewroÔme thn

α

dx ≡ β

dmod

m

d.Aut èqei mia monadik lÔsh thn x0 ≡

(α

d

)∗ β

dmod

m

dìpou (α

d

)∗ e�nai oant�strofo tou α

dmod

m

dpou sun jw upolog�zetai me ton Eukle�deioAlgìrijmo. An up�rqei o p�naka twn deikt¸n mod

m

d, tìte h eÔresh th lÔsh e�nai �mesh. Pr�gmati, èqoume ind

(α

dx)

≡(

indβ

d

)

modϕ(m

d

)

indx ≡(

indβ

d− ind

α

d

)

modϕ(m

d

)

.Opìte apì ton p�naka twn deikt¸n modm

dbr�skoume to x0 pou èqei de�kth

(

indβ

d− ind

α

d

)

modϕ(m

d

).

3.7. Diwnumikè Isotim�e 175'Eqonta brei autì to x0, br�skoume ti upìloipe d − 1 mh-isìtime lÔ-sei th αx ≡ βmodm pou w gnwstì e�nai oi x0 +m

d, . . . , x0 + (d −

1)m

dmodm.Par�deigma. Na luje� h ex�swsh 19x ≡ 17mod 26. Epeid m = 2 · 13,up�rqoun prwtarqikè r�ze mod 26. Epeid o 2 e�nai mia �rtia prwtarqik r�za

mod 13 sÔmfwna me to 3.5.9, o 2 + 13 = 15 e�nai mia prwtarqik r�za mod26.JewroÔme tou de�kte mod 26 w pro th b�sh 15. 'Eqoumeindx ≡ (ind17 − ind19)mod 12.All� sÔmfwna me ton p�nakaOi timè tou ind15α (mod 26)

α: 1 3 5 7 9 11 15 17 19 21 25indα: 0 4 9 11 8 7 1 2 5 3 6èqoume indx ≡ 2− 5 ≡ −3 ≡ 9mod 12. 'Ara, p�li apì ton p�naka br�skoume ìti

x = 5, pou pr�gmati 5 · 19 = 95 ≡ 17mod 26.3.7 Diwnumikè Isotim�e Mia poluwnumik isotim�aαxn ≡ βmodmonom�zetai diwnumik isotim�a. H ep�lush aut¸n, ìpw èqoume dei, an�getai sthnep�lush diwnumik¸n isotimi¸n mod p, p pr¸to . Sunep¸ arke� na melet soumethn per�ptwshαxn ≡ βmod p,ìpou p e�nai pr¸to kai p ∤ α. Pollaplasi�zonta ep� ton ant�strofo α∗ tou

αmod p, pa�rnoume thnxn ≡ βα∗ mod p,

176 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze pou èqei ti �die lÔsei me thn prohgoÔmenh. 'Etsi telik� arke� na melet soumethn per�ptwshxn ≡ αmod p, p pr¸to .An α ≡ 0mod p, tìte oi lÔsei e�nai oi x = κp, κ ∈ Z. 'Estw α 6≡ 0mod p. Giana up�rqoun lÔsei th (∗) prèpei kai arke�(**) nindgx ≡ indgαmod p− 1ìpou g e�nai opoiad pote prwtarqik r�za mod p.Diakr�noume dÔo peript¸sei . 'Estw δ = (n, p− 1).1η per�ptwsh: indgα ≡ 0mod δ. Tìte h (∗∗) èqei akrib¸ δ mh-isìtime lÔsei

mod p− 1 kai �ra kai h (∗) èqei δ mh-isìtime lÔsei mod p. Qrhsimopoi¸nta to Je¸rhma 3.6.1, h sunj kh indgα ≡ 0mod δ isodÔnama h p− 1

δindgα ≡

0mod

(

p− 1

δδ

) e�nai isodÔnamh me thn indgαp−1

δ ≡ 0mod p − 1 αp−1

δ ≡

1mod p. Dhlad h (∗) èqei lÔsh an kai mìnon an α p−1δ ≡ 1mod p. (Autì e�naito krit rio tou Euler sto opo�o ja anaferjoÔme sto epìmeno kef�laio).2η per�ptwsh: indgα 6≡ 0mod δ. S� aut thn per�ptwsh e�nai gnwstì ìti h (∗∗)den èqei kam�a lÔsh, opìte to �dio isqÔei kai gia thn (∗).Parade�gmata.1. 'Estw p perittì pr¸to kai g mia prwtarqik r�za mod p. 'Estw α, 1 ≤

α ≤ p−1, tìte h x2 ≡ αmod p èqei lÔsh an kai mìnon an 2|indgα. An aut h sunj kh ikanopoie�tai tìte up�rqoun akrib¸ dÔo mh-isìtime lÔsei . Oidun�mei gκ, 0 ≤ κ ≤ p− 2 e�nai isìtime me tou akèraiou 1, 2, . . . , p− 1(me k�poia 1 − 1 antistoiq�a). Oi �rtie dun�mei gκ, 0 ≤ κ ≤ p − 2 ma d�noun ti timè tou α gia ti opo�e h isotim�a x2 ≡ αmod p e�nai epilÔsimh,afoÔ (gκ)p−12 ≡ 1mod p an kai mìnon an p − 1|κ

(

p− 1

2

) pou shma�neiìti o κ e�nai �rtio . Up�rqoun de p− 1

2tètoie epilogè gia to α. Giapar�deigma, an p = 13, oi timè tou α, gia g = 2, e�nai g0, g2, g4, g6, g8, g10,

3.7. Diwnumikè Isotim�e 177dhlad 1, 4, 3, 12, 9, 10 ant�stoiqa. Autì shma�nei ìti mìno gi� autè ti timè tou α h x2 ≡ αmod13 èqei lÔsei . Gia α = 1, 4, 3, 12, 9, 10 oiant�stoiqe lÔsei e�nai ±1, ±2, ±4, ±8, ±3, ±6 ant�stoiqa.2. Na luje� h 7x4 ≡ 11mod 26. LÔnoume ti 7x4 ≡ 11mod 13 kai 7x4 ≡11mod 2. Autè èqoun ti �die lÔsei me ti x4 ≡ 9mod 13 kai x4 ≡1mod 2. Gia thn pr¸th, èqoume δ = (4, 12) = 4 kai α = 9 kai epeid ind9 = 8 kai 4|8 ja èqoume δ = 4 mh isìtime lÔsei (autì bèbaia prokÔpteikai apì to 3.3.5). Gia na ti broÔme lÔnoume thn 4indx ≡ 8mod 12. Apìthn indx ≡ 2mod 3 pa�rnoume thn indx0 = 2mod 12 ap� aut indx ≡5, 8, 11mod 12. Opìte x0 = 4 kai x = 6, 9, 7mod 13. Gia th deÔterh hmình lÔsh e�nai profan¸ h 1mod 2. 'Etsi èqoume ta 4 sust mata

x ≡ 1mod 2

x ≡ 4mod 13

x ≡ 1mod 2

x ≡ 6mod 13

x ≡ 1mod 2

x ≡ 9mod 13

x ≡ 1mod 2

x ≡ 7mod 13pou èqoun ant�stoiqa ti monadikè lÔsei 17, 19, 22, 20mod 26.3. Na luje� h ex�swsh x3 ≡ αmod17 gia ìla ta α ∈ Z. MporoÔme naupojèsoume ìti to α e�nai metaxÔ tou 0 kai tou 16. An α = 0, h x ≡0mod 17 e�nai h monadik lÔsh. 'Estw ìti α 6= 0. Br�skoume ìti to 3 e�naimia prwtarqik r�za mod17 kai sunep¸ to α e�nai mia dÔnamh 3κ tou 3gia k�poio κ, 0 ≤ κ ≤ 15. Qrhsimopoi¸nta th sqèsh

ind(n− α) ≡ ind(−α) ≡(

1

2ϕ(n) + indα

)

modϕ(n),dhlad sth prokeimènh per�ptwsh: ind(17−α) ≡ (8+ indα)mod 16 pa�r-noume ton p�naka deikt¸n:P�naka deikt¸n ind3αmod17

α: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

indα: 0 14 1 12 5 15 11 10 2 3 7 13 4 9 6 8Kaj¸ to (3, 16) = 1|ind3α = κ, h ex�swsh x3 ≡ αmod 17 èqei mìno mialÔsh mod 17. Aut h lÔsh ja e�nai th morf gλ, 0 ≤ λ ≤ 15, ìpou to λ

178 Kef�laio 3. Poluwnumikè Isotim�e kai Prwtarqikè R�ze upolog�zetai apì thn ex�swsh 3λ ≡ κmod 16 kai e�nai λ = 11 · κmod 16,afoÔ 3 · 11 ≡ 33 ≡ 1mod 16.Gia α =1, 2, 3, 4, 5, 6, 7 kai 8 (apì ton p�naka br�skoume to κ = indα kaimet� to λ) h ant�stoiqh (monadik ) lÔsh th x3 ≡ αmod17 e�nai h 3λ ≡1,8, 7, 13, 11, 5, 14 kai 2. (Gia par�deigma, an α = 6, èqoume κ = ind6 = 15,opìte λ ≡ 11 · 15 = 165 ≡ 5mod 16 kai 35 ≡ 5mod 17. 'Ara (35)3 ≡ 53 ≡6mod 17. Dhlad h lÔsh th x3 ≡ 6mod 17 e�nai 5mod 17). Epeid de anβ3 ≡ αmod17, 1 ≤ β, α ≤ 16, tìte kai (17− β)3 ≡ (17−α)mod 17, giaα = 9,10, 11, 12, 13, 14, 15 kai 16 h ant�stoiqh lÔsh th x3 ≡ αmod17e�nai 15, 3, 12, 6, 4, 10, 9 kai 16.An jèlame na lÔsoume thn x3 ≡ αmod 172, epeid f ′(x) = 3x2, ìpouf(x) = x3 − α, me (3, 17) = (β2, 17) = (3β2, 17) = 1, gia k�je β, 1 ≤β ≤ 16, e�nai f ′(β) 6≡ 0mod 17. Sunep¸ sÔmfwna me to 3.4.3, h x3 ≡αmod172 èqei th monadik lÔsh (β + 17 · t)mod 172, ìpou β e�nai lÔshth x3 ≡ αmod17 kai t ≡ −f ′(β)

f(β)

pmod p. Gia par�deigma gia α = 6tìte β = 5 kai t = −1mod p, dhlad h 5 − 17 ≡ −12mod 172 e�nai lÔshth x3 ≡ 6mod 172. Me ton �dio trìpo mporoÔme na broÔme ti lÔsei th

x3 ≡ αmod17n gia k�je n ≥ 1.Parat rhsh. Sto Par�deigma 1 e�dame ìti ìle oi �rtie dun�mei th prw-tarqik r�za gmod p ma d�noun ti timè tou α gia ti opo�e h isotim�ax2 ≡ αmod p (p perittì pr¸to ) e�nai epilÔsimh me p ∤ α. Up�rqoun de p− 1

2tètoie epilogè gia to α, dhlad up�rqoun p− 1

2�rtie dun�mei tou g kaiautè e�nai oi g0, g2, g4, . . . , g2( p−3

2 ). To �dio e�dame kai sto Par�deigma 3, ìpous� autì, ìle oi dun�mei 3λ (se pl jo p − 1 = 16) ma d�noun ti timè touα gia ti opo�e h isotim�a x3 ≡ αmod17 e�nai epilÔsimh (me p ∤ α). Autì dene�nai tuqa�o. An h ex�swsh xn ≡ αmodm èqei lÔsh tìte lème ìti α e�nai mian-iost dÔnamh modm. MporoÔme den na jewr soume to α metaxÔ tou 0 kaitou m− 1. To 0 kai to 1 e�nai p�nta n-iostè dun�mei modm gia k�je m kain, afoÔ on ≡ 0modn kai 1n ≡ 1modm. JewroÔme ìti o m e�nai èna pr¸to arijmì p. Prohgoumènw e�dame ìti h xn ≡ αmod p, p ∤ α, èqei lÔsh an kai

3.7. Diwnumikè Isotim�e 179mìnon anα

p−1δ ≡ 1mod pìpou δ = (n, p − 1) isodÔnama an kai mìnon an δ|indgα, ìpou g e�nai miaprwtarqik r�za mod p. Dhlad h xn ≡ αmod p èqei lÔsh an kai mìnon an o αe�nai lÔsh th

xp−1

δ ≡ 1mod p(pou aut èqei akrib¸ p− 1

δmh isìtime lÔsei , sÔmfwna me to 3.3.6) iso-dÔnama an kai mìnon an o α e�nai isìtimo mod p me mia apì ti p− 1

δdun�-mei g0, gδ, g2δ , . . . , g(

p−1−δδ )δ (afoÔ δ|indα ≤ p − 2 an kai mìno an indα =

0, δ, 2δ, . . . ,

(

p− 1 − δ

δ

)

δ, me 0 ≤ indα ≤ p − 2). Sunep¸ up�rqoun p− 1

δmh-isìtime n-iostè dun�mei (dhlad epilogè gia to α)mod p.Ep�sh , e�nai fanerì, an jewr soume to sÔnolo {1, g, . . . , gp−2} pou e�-nai pl re sÔsthma antipros¸pwn pr¸twn kl�sewn mod p, tìte to sÔnolo{1n, gn, . . . , gn(p−2)} perièqei ìle ti n-iostè dun�mei mod p pou e�nai pr¸-te pro to p. K�je de n-iost dÔnamh mèsa s� autì to sÔnolo emfan�zetai δforè afoÔ ìpw èqoume dei, gia dosmèno α, p ∤ α, xn ≡ αmod p èqei akri-b¸ δ mh-isìtime lÔsei . To �dio ja isqÔei kai gia to sÔnolo twn dun�mewn{αn

1 , αn2 , . . . , α

np−1} an to {α1, . . . , αp−1} e�nai èna opoiod pote pl re sÔsthmaantipros¸pwn pr¸twn kl�sewn upolo�pwn mod p. Gia par�deigma, gia p = 13ta tetr�gwna {12, 22, (22)2, (23)2, . . . , (211)2} perièqoun ìle ti 2-iostè dun�-mei mod13 pou (ìpw e�dame) e�nai oi (12

2= 6 to pl jo ) 1, 3, 4, 9, 10 kai12mod 13 kai k�je m�a emfan�zetai dÔo forè : to 1 w 12 kai w (26)2, to 3 w

(22)2 kai w (28)2, to 4 w 22 kai w (27)2, to 9 w (24)2 kai w (110)2, to 10w (25)2 kai w (211)2 kai to 12 w (23)2 kai w (29)2.

Kef�laio 4Tetragwnik� Upìloipa kai oNìmo Antistrof 4.1 Tetr�gwna mod pSto prohgoÔmeno Kef�laio e�dame ìti to prìblhma th melèth twn poluwnumi-k¸n isotimi¸n th morf

f(x) ≡ 0modmìpou f(x) ∈ Z[x] kai m ∈ N, m > 1, an�getai sto prìblhma th melèth twnpoluwnumik¸n isotimi¸n th morf f(x) ≡ 0mod pìpou f(x) ∈ Z[x] kai p e�nai èna pr¸to arijmì . Gi� autè ti poluwnumikè isotim�e mod p, an kai anafèrjhkan orismèna apotelèsmata pou bohjoÔn sthnep�lus tou , de dìjhke kam�a genik mèjodo gia thn eÔresh twn lÔse¸n tou ,oÔte apant jhke to er¸thma an up�rqoun den up�rqoun lÔsei mia tètoia genik isotim�a . Autì e�nai èna apì ta pio shmantik� probl mata th Jewr�a Arijm¸n pou mèqri s mera paramènei anap�nthto.Sthn eidik per�ptwsh pou to polu¸numo f(x) e�nai pr¸tou bajmoÔ, f(x) =

αx + β, p ∤ α, e�dame ìti h f(x) ≡ 0mod p èqei th monadik lÔsh x =181

182 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof ap−2βmod p an p ∤ β kai fusik� èqei mìno th mhdenik an p|β. Sto kef�laioautì ja melet soume diexodik� th deutèrou bajmoÔ (dutetr�gwnh) isotim�a(1) α2x

2 + α1x+ α0 ≡ 0mod p, p ∤ α2.'Opw ja doÔme, gia autè ti exis¸sei , to prìblhma th Ôparxh mh Ôparxh lÔsewn èqei luje� pl rw . 'Omw akìma kai s� aut thn per�ptwsh den up�rqeimia genik mèjodo pou na ma odhge� sthn eÔresh twn lÔsewn (ìtan up�rqoun).An p = 2, epeid oi pijanè lÔsei th (1) e�nai 0 kai 1 aut p�nta mpore�na luje� (gia perittoÔ a1 kai α0 den èqei lÔsh en¸ gia �rtiou α1 kai α0 èqeimìno th mhdenik . Sti �lle peript¸sei èqei th monadik lÔsh 1). Sunep¸ sto ex mporoÔme na upojètoume ìti p 6= 2.E�nai gnwstì ìti k�je migadikì arijmì e�nai to tetr�gwno enì migadikoÔarijmoÔ en¸ èna pragmatikì arijmì e�nai to tetr�gwno enì pragmatikoÔarijmoÔ an kai mìnon an autì e�nai megalÔtero �so apì to mhdèn. Togegonì autì, dhlad h gn¸sh th fÔsh twn tetrag¸nwn gia tou migadikoÔ kai tou pragmatikoÔ arijmoÔ , e�nai kajoristikì gia thn Ôparxh mh Ôparxhmigadik¸n pragmatik¸n lÔsewn th genik algebrik ex�swsh (2) α2x2 + α1x+ α0 = 0, α2 6= 0.Pr�gmati e�nai gnwstì, apì thn �lgebra tou sqole�ou, ìti oi lÔsei th (2)d�dontai apì ton tÔpo(3) x =

−α1 ±√

α21 − 4α2α0

2α2.Sunep¸ h (2) me α2, α1, α0 migadikoÔ èqei p�nta lÔsei migadikè afoÔ to α2

1−4α2α0 e�nai p�nta to tetr�gwno enì migadikoÔ, en¸ me α2, α1, α0 pragmatikoÔ h (2) èqei pragmatikè lÔsei an kai mìnon an o α2

1 − 4α2α0 e�nai to tetr�gwnoenì pragmatikoÔ arijmoÔ, dhlad an kai mìno an α21 − 4α2α0 ≥ 0. To �dioisqÔei kai gia thn (1). Pr�gmati, to tÔpo (3) prokÔptei pollaplasi�zonta th(2) ep� 4α2 kai parathrìnta ìti

4α2(α2x2 + α1x+ α0) = (2α2x+ α1)

2 − (α21 − 4α2α0).

4.1. Tetr�gwna mod p 183To �dio mporoÔme na efarmìsoume sthn (1). Epeid 4α2 6≡ 0mod p, h (1) èqeiti �die lÔsei me thn4α2(α2x

2 + α1x+ α0) = (2α2x+ α1)2 − (α2

1 − 4α2α0) ≡ 0mod p.Sunep¸ h (1) èqei lÔsh an kai mìnon an o (α21−4α2α0) e�nai isìtimo modulo pme to tetr�gwno k�poiou akèraiou arijmoÔ (pou mpore� na periorisje� metaxÔ tou0 kai p− 1). Sunep¸ to prìblhma th ep�lush th (1) an�getai sto prìblhmaep�lush th ex�swsh (4) x2 ≡ αmod p.An p|α, tìte aut èqei mìno th lÔsh x ≡ 0mod p.Sto ex ja upojètoume ìti p ∤ α. Apì to Je¸rhma tou Lagrange h (4) èqeito polÔ dÔo lÔsei mod p. An x0 e�nai mia lÔsh tìte kai h −x0 = p − x0 e�nailÔsh di�forh mod p th x0, afoÔ diaforetik� ja e�qame 2x0 ≡ 0mod p kai �ra

x0 ≡ 0mod p (afoÔ p 6= 2) pou e�nai �topo (apì thn upìjesh p ∤ a). Sunep¸ h(4) èqei dÔo mh-isìtime mod p lÔsei den èqei kam�a lÔsh (de kai Kef�laio3, sel. 143).Sthn per�ptwsh pou h (4) èqei lÔsh tìte lème ìti o akèraio α e�nai ènatetragwnikì upìloipo mod p ( TU en suntom�a). An h (4) den èqei lÔsh tìtelème ìti o α e�nai èna tetragwnikì mh-upìloipo ( TMU en suntom�a) mod p.A doÔme orismèna parade�gmata. Ston epìmeno p�naka katagr�foume tatetr�gwna kai ta mh tetr�gwna 6≡ 0mod p gia tou pr¸tou p = 3, 5, 7, 11, 13, 17kai 19.Pr¸toi Tetr�gwna mod p Mh tetr�gwna mod p3 1 25 1, 4 2, 37 1, 2, 4 3, 5, 611 1, 3, 4, 5, 9 2, 6, 7, 8, 1013 1, 3, 4, 9, 10, 12 2, 5, 6, 7, 8, 1117 1, 2, 4, 8, 9, 13, 15, 16 3, 5, 6, 7, 10, 11, 12, 1419 1, 4, 5, 6, 7, 9, 11, 16, 17 2, 3, 8, 10, 12, 13, 14, 15, 18

184 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof S� autì ton p�naka, e�nai dÔskolo na doÔme an up�rqei k�poia kanonikìth-ta sthn emf�nish twn tetrag¸nwn oÔtw ¸ste na mporoÔme na eik�soume ènakrit rio to opo�o na ma lèei pìte èna akèraio e�nai den e�nai èna TU mod p.Fusik� to 1 e�nai p�nta èna tetr�gwno. All� sto er¸thma pìte to 2 e�naièna TU mod p den e�nai tìso eÔkolo na apant soume. Gia tou akèraiou pouan koun sthn kl�sh p − 1 ≡ −1mod p ja mporoÔsame na eik�soume ìti auto�e�nai TU mod p an kai mìnon an p ≡ 1mod 4, afoÔ o p− 1 e�nai TU mod p giap = 5, 13 kai 17 en¸ den e�nai gia p = 3, 7, 11 kai 19. 'Hdh ìmw gnwr�zoume apìto Je¸rhma 2.3.12 ìti aut h eikas�a e�nai alhj . Aut ìmw e�nai ep�sh �meshsunèpeia tou krithr�ou tou Euler pou anafèrjhke sto tèlo tou Kef�laiou 3.T¸ra diatup¸noume autì to krit rio se mia pio pl rh morf gia ta TU.4.1.1 Je¸rhma (Krit rio tou Euler). 'Estw α ∈ Z me p ∤ α. Tìte

i) o α e�nai èna TU mod p an kai mìnon an isqÔeiαp−1/2 ≡ +1mod p

ii) o α e�nai èna TMU mod p an kai mìnon an isqÔeiαp−1/2 ≡ −1mod p.Apìdeixh 1η. i) Autì èqei deiqje� w efarmog twn prwtarqik¸n riz¸n stotèlo tou Kefala�ou 3 gia δ = 2.

ii) Oi lÔsei th x2 ≡ 1mod p e�nai h 1mod p kai h −1mod p. Apì toJe¸rhma tou Fermat isqÔei(αp−1/2)2 = αp−1 ≡ 1mod p.Dhlad h kl�sh αp−1/2 mod p e�nai lÔsh th x2 ≡ 1mod p kai sunep¸ jaèqoume αp−1/2 ≡ 1mod p ja èqoume αp−1/2 ≡ −1mod p. All� apì to i) o

α e�nai TMU an kai mìnon an αp−1/2 6≡ 1mod p kai �ra o α e�nai TMU an kaimìnon an αp−1/2 ≡ −1mod p.

4.1. Tetr�gwna mod p 185Apìdeixh 2η. Ed¸ den ja qrhsimopoi soume thn ènnoia th prwtarqik r�za all� th dom tou sunìlou {1, 2, . . . , p − 1} w pro ton pollaplasiasmì twnstoiqe�wn tou mod p.i) JewroÔme ta tetr�gwna 12, 22, . . . ,

(p− 1

2

)2, . . . , (p − 1)2. K�je èna tè-toio tetr�gwno e�nai isìtimo mod p me èna apì ta tetr�gwna 12, 22, . . . ,

(p− 1

2

)2.Pr�gmati, an κ = 1, 2, . . . ,p− 1

2, tìte κ2 = (−κ)2 = (p − κ)2. Ep�sh ìla ta tetr�gwna 12, 22, . . . ,

(p− 1

2

)2 e�nai an� dÔo mh-isìtima, afoÔ an 1 ≤

κ1 < κ2 ≤ p− 1

2, tìte κ2

1 ≡ κ22 mod p an kai mìnon an κ2 − κ1 ≡ 0mod p

κ1 + κ2 ≡ 0mod p, all� 1 ≤ κ2 − κ1 ≤ p− 1

2kai 3 ≤ κ1 + κ2 ≤ p − 1, opìteden mpore� na isqÔei oÔte κ2 − κ1 ≡ 0mod p oÔte κ1 + κ2 ≡ 0mod p. Sunep¸ k�je TU α e�nai isìtimo mod p me èna ap� aut� ta tetr�gwna, dhlad ja èqoume

α ≡ κ2 mod p, ìpou κ = 1, 2, . . . ,p− 1

2. Opìte apì to Je¸rhma tou Fermatpa�rnoume

αp−12 ≡ κ2( p−1

2 ) = κp−1 ≡ 1mod p.Ant�strofa, ìti isqÔei α p−12 ≡ 1mod p. Ja de�xoume ìti o α e�nai èna TU.Autì e�nai isodÔnamo me to na de�xoume ìti an o α den e�nai TU, dhlad e�naièna TMU, tìte isqÔei αp−1/2 ≡ −1mod p. Upojètoume loipìn ìti o α e�naièna TMU. Tìte gia k�je i ∈ {1, 2, . . . , p− 1} up�rqei èna j ∈ {1, 2, . . . , p− 1}tètoio ¸ste ij ≡ αmod p. Pr�gmati, an i ∈ {1, 2, . . . , p − 1}, up�rqei èna

i∗ ∈ {1, 2, . . . , p − 1}, to ant�strofo tou i modulo p, opìte i(i∗α) ≡ αmod p.Jètoume i∗α ≡ jmod p, j ∈ {1, 2, . . . , p−1} kai èqoume ij ≡ αmod p. EpiplèonisqÔei i 6≡ jmod p, diìti diaforetik� ja e�qame i2 ≡ αmod p opìte to α den ja tan TMU. 'Etsi èqoume diamer�sei to sÔnolo {1, 2, . . . , p−1} se zeug�ria (i, j)me ij ≡ αmod p kai i 6≡ jmod p. Up�rqoun de akrib¸ p− 1

2diaforetik� tètoiazeug�ria. Pollaplasi�zonta ìle maz� ti ant�stoiqe isotim�e ij ≡ mod ppa�rnoume: 1 · 2 · 3 · · · (p− 1) ≡ α

p−12 mod p. All� apì to Je¸rhma tou WilsonprokÔptei ìti −1 ≡ (p− 1)! = 1 · 2 · · · (p− 1) ≡ α

p−12 mod p.Parathr sei .

186 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof 1. H 2η apìdeixh tou 4.1.1 e�nai mia gen�keush th idèa pou diatup¸netaisthn Parat rhsh 2 met� to Je¸rhma 2.3.12 kai antistoiqe� sthn eidik per�ptwsh α = −1.2. Sthn 2η apìdeixh ep�sh e�dame ìti to pl jo twn TU isoÔtai me to pl jo twn TMU, dhlad p− 1

2. Ta TU e�nai isìtima me tou akèraiou 12, 22, . . . ,

(p− 1

2

)2, en¸ ta TMU e�nai isìtima me tou upìloipou akèraiou metaxÔtou 1 kai p− 1 pou den e�nai isìtima me tou 12, 22, . . . ,(p− 1

2

)2.3. (Ginìmeno kai 'Ajroisma TU). Epeid p− κ ≡ −κmod p, èqoume−1 ≡ (p− 1)! = 1 · 2 · · · p− 1

2

p+ 1

2· · · (p− 1)

= 1 · 2 · · · p− 1

2

(

p− p− 1

2

)

· · · (p − 1)

≡ 1 · 2 · · ·p− 1

2

(

− p− 1

2

)(

− p− 3

2

)

· · · (−1)

=(

1 · 2 · · · p− 1

2

)

(−1)p−12

(

1 · 2 · · · p− 1

2

)

= (−1)p−12

((p− 1

2

)

!)2

mod p.Opìte 12 · 22 · · ·(p− 1

2

)2≡ (−1)

p+12 mod p. Ep�sh epeid o p e�nai pe-rittì , o p−κ e�nai �rtio ( ant�stoiqa perittì ) an kai mìnon an o κ e�naiperittì (ant�stoiqa �rtio ) kai �ra sto ginìmeno

1 · 2 · · · (p− 1)to pl jo twn �rtiwn paragìntwn isoÔtai me to pl jo twn peritt¸nparagìntwn (ki autì to pl jo e�nai p− 1

2

). 'Eqoume de−1 ≡ 1 · 2 · · · (p − 1) = 2(p− 2)4(p − 4)6(p − 6) · · · (p − 1)(p − (p− 1))

≡ (−22)(−42)(−62) · · · − (p− 1)2) = (−1)p−12 22 · 42 · · · (p− 1)2

≡ 1(p− 1)3(p − 3)5(p − 5) · · · (p − 2)(p − (p − 2))

≡ (−1)p−12 12 · 32 · 52 · · · (p − 2)2 mod p.

4.1. Tetr�gwna mod p 187Sunep¸ pa�rnoume((p− 1

2

)

!)2

= 12 · 22 · 32 · · ·(p− 1

2

)2≡ 22 · 42 · · · (p − 1)2

≡ 12 · 32 · 52 · · · (p− 2)2 ≡ (−1)p+12 mod p

≡

1 an p+ 1

2≡ 0mod 2

−1 an p+ 1

2≡ 1mod 2.Aut thn isotim�a ep�sh mporoÔme na thn apode�xoume qrhsimopoi¸nta prwtarqikè r�ze mod p w ex : 'Estw g mia prwtarqik r�za, tìte gnw-r�zoume ìti èna tetragwnikì upìloipo α e�nai isìtimo me mia �rtia dÔnamh

g2λ tou g, 1 ≤ λ ≤ p− 1/2. 'Etsi an α1, α2, . . . , α p−12

e�nai ta TU mod psto di�sthma 1 ≤ αi ≤ p− 1, tìteα1α2 · · ·α p−1

2≡ g2g4 · · · g2 p−1

2 mod p ≡ g2(1+2+···+ p−12 ) mod p

≡ g2( p−12

p+12 )/2 ≡ g

p−12

p+12 ≡ (−1)

p+12 mod p

≡

1 an p+ 1

2≡ 0mod p

−1 an p+ 1

2≡ 1mod p

≡{

1 an p ≡ −1mod 4

−1 an p ≡ 1mod 4.Gnwr�zoume ìti èna tetragwnikì upìloipo emfan�zetai (mod p) akrib¸ dÔo forè metaxÔ twn 12, 22, . . . ,(

p−12

)2, . . . , (p − 1)2. Sunep¸

12 + 22 + · · · + (p − 1)2 ≡ 2(

12 + 22 + · · · +(p− 1

2

)2)

mod p.All�12 + 22 + · · · + (p− 1)2 =

1

6(p− 1)p(2p − 1).Opìte an p = 3, tìte 12 + 22 ≡ −1mod 3.An p > 3, tìte 12 + 22 + · · · +

(p− 1

2

)2≡ 0mod p.Autì prokÔptei �mesa kai apì thn ex endiafèrousa sqèsh. Gia k�je

188 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof n ∈ N isqÔeiSn(p) = 1n + 2n + 3n + · · · + (p − 1)n ≡

{

−1mod p an p− 1|n0mod p an p− 1 ∤ n.Pr�gmati, èstw ìti n = (p − 1)m. Tìte

Sn(p) =1(p−1)m + · · · + (p− 1)(p−1)m

=(1p−1)m + (2p−1)m + · · · + ((p − 1)p−1)m

≡ (1 + 1 + · · · + 1)mod p ≡ (p− 1)mod p ≡ −1mod p.'Estw ìti p − 1 ∤ n. Oi akèraioi 1, 2, . . . , p − 1 e�nai oi akèraioi (�sw mediaforetik di�taxh) g, g2, . . . , gp−1 modulo p. 'AraSn(p) ≡ gn + (g2)n + · · · + (gp−1)n ≡ gn + (gn)2 + · · · + (gn)p−1 mod p

≡ gn(1 + gn + · · · + (gn)p−2 mod p

≡ gn(gn − 1)∗((gn)p−1 − 1)mod p ≡ 0mod pìpou (gn − 1)∗ mod p e�nai h ant�strofh kl�sh th (gn − 1)mod p (pouup�rqei afoÔ h t�xh p − 1 tou g de diaire� to n kai �ra gn 6≡ 1mod p,dhlad gn − 1 6≡ 0mod p).4. An α e�nai èna TMU tìte ìla ta TMU e�nai isìtima me tou akèraiou α12, α22, . . . , α

(

p−12

)2. Pr�gmati, auto� oi akèraioi e�nai an� dÔo mh-isìtimoi mod p kai èqoume(ακ2)

p−12 = α

p−12 κp−1 ≡ α

p−12 ≡ −1mod pìpou κ e�nai èna apì tou 1, 2, . . . ,

p− 1

2.6. 'Otan o pr¸to arijmì p e�nai sqetik� mikrì arijmì , tìte to Krit -rio tou Euler ma bohj� stou upologismoÔ gia na exet�soume an èna akèraio α e�nai den e�nai TU mod p. Gia par�deigma, an p = 29, tìteèqoume gia ton α = 2

229−1

2 = 214 = (27)2 = (2522)2 ≡ (3 · 4)2 ≡ 28 ≡ −1mod29

4.2. To SÔmbolo tou Legendre 189kai �ra o 2 den e�nai TU mod29. An ìmw o p e�nai arket� meg�lo tìteme thn efarmog tou Krithr�ou tou Euler prokÔptoun upologismo� poue�nai arket� qronobìroi. Pio k�tw ja anaptÔxoume perissìtero apotele-smatikè mejìdou gia tètoiou upologismoÔ . Pollè forè to Krit riotou Euler qrhsimopoie�tai se jewrhtik� probl mata. Gia par�deigma ap�autì to Krit rio èqoume4.1.2 Pìrisma. O akèraio −1 e�nai TU an kai mìnon an p ≡ 1mod 4.Apìdeixh. O −1 e�nai TU an kai mìnon an (−1)p−12 ≡ 1mod p dhlad an kaimìnon an p− 1

2= 2n, pou shma�nei ìti p = 1 + 4n.4.1.3 Pìrisma. An o p e�nai èna pr¸to diairèth enì akèraiou th morf

n2 + 1 tìte p ≡ 1mod 4.Apìdeixh. Epeid p|n2 + 1, dhlad n2 ≡ −1mod p, o −1 e�nai TU mod p kai�ra, apì to 4.1.2, p ≡ 1mod 4.4.1.4 Pìrisma. 'Estw p èna pr¸to arijmì tètoio ¸ste p ≡ 1mod 4. Anα e�nai èna TMU mod p, tìte o α p−1

4 e�nai lÔsh th x2 ≡ −1mod p.Apìdeixh. Apì to Krit rio tou Euler èqoume α p−12 ≡ −1mod p, dhlad (

αp−14

)2=

α2( p−14 ) ≡ −1mod p.4.2 To SÔmbolo tou LegendreO Adrien Marie Legendre (1752�1833) eis gage ton epìmeno sumbolismì o opo�-o èqei kajierwje� na qrhsimopoie�tai sth Jewr�a Arijm¸n giat� aplousteÔeitou upologismoÔ kai ti diatup¸sei apotelesm�twn pou anafèrontai sta TUkai TMU.'Estw p èna pr¸to perittì arijmì kai α èna akèraio arijmì . To

190 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof sÔmbolo tou Legendre (α/p) tou αmod p or�zetai w ex :(α/p) =

1 an o α e�nai èna TU mod p

−1 an o α e�nai èna TMU mod p

0 an p|α.Gia par�deigma mporoÔme na gr�youme ìti to pl jo twn lÔsewn th x2 ≡ αmod pe�nai �so me 1 + (α/p). Ep�sh to Krit rio tou Euler mpore� na diatupwje� w :

(α/p) ≡ αp−1/2 mod p kai to Pìrisma 4.1.2 w (−1/p) = (−1)p−1/2.Sto ex ja suneq�soume na upojètoume ìti ìle oi kl�sei upolo�pwnmod p pou ja melet�me e�nai antistrèyime , opìte to sÔmbolo tou Legendreja e�nai 6= 0.'Ena apì tou lìgou pou o Legendre eis gage to sÔmbolo (α/p) e�nai oex : Gnwr�zoume, apì to Je¸rhma 3.6.1 i) ìti �n α, β ∈ Z, (α, p) = (β, p) = 1,tìte

ind (αβ) ≡ ( indα+ indβ)mod (p− 1).Epeid o p− 1 e�nai �rtio pa�rnoume thn isotim�aind (αβ) ≡ ( indα+ indβ)mod 2.Diakr�noume trei peript¸sei :1h per�ptwsh: an o α kai o β e�nai TU mod p, tìte oi de�kte indα kai indβe�nai �rtioi arijmo� (afoÔ o α kai o β e�nai �rtie dun�mei th prwtarqik r�za

mod p pou jewroÔme w b�sh gia tou de�kte ). 'Araindαβ ≡ indα+ indβ ≡ 0 + 0 ≡ 0mod 2.Dhlad s� aut thn per�ptwsh to ginìmeno αβ e�nai TU mod p. Me �lla lìgia,sumbolik�, èqoume

TY × TY = TY.

4.2. To SÔmbolo tou Legendre 1912h per�ptwsh: an o α e�nai TU kai o β e�nai TMU, tìte indα ≡ 0mod 2 kaiindβ ≡ 1mod 2 kai sunep¸ indαβ ≡ 0 + 1 ≡ 1mod 2. Dhlad , o αβ e�naiTMU. 'Ara

TY × TMY = TMY.

3h per�ptwsh: an o α kai o β e�nai TMU. Tìte, epeid indα ≡ 1mod 2 kaiindβ ≡ 1mod 2, èqoume ind (αβ) ≡ 1 + 1 ≡ 0mod 2. Dhlad o αβ e�nai TU.'Ara, sumbolik� èqoume

TMY × TMY = TY.Autè oi trei dunatè peript¸sei tou pollaplasiasmoÔ twn TU kai TMUma upenjum�oun ti trei dunatè peript¸sei pollaplasiasmoÔ tou 1 kai −1,dhlad ìti ant�stoiqa èqoume 1 × 1 = 1, 1 × (−1) = −1 kai (−1) × (−1) = 1.Qrhsimopoi¸nta to sÔmbolo tou Legendre èqoume4.2.1 L mma. Gia dÔo antistrèyime kl�sei αmod p kai βmod p isqÔei(α/p)(β/p) = (αβ/p).Eidik� isqÔei

(α2/p) = 1 kai (α2β/p) = (β/p).Apìdeixh. ProkÔptei �mesa apì thn prohgoÔmenh parat rhsh.4.2.2 Pìrisma. 'Estw α ∈ N me p ∤ α. Tìtekai (α/p) = (p1/p)n1(p2/p)

n2 · · · (ps/p)ns

(−α/p) = (−1)p−1/2(α/p),ìpou α = pn11 · · · pns

s e�nai h an�lush tou α se pr¸tou arijmoÔ .Parade�gmata.1. Na deiqje� ìti p−1∑

α=1(α/p) = 0. Pr�gmati, gnwr�zoume ìti up�rqoun p− 1

2TU kai p− 1

2TMU, opìte

p−1∑

α=1

(α/p) =∑

αTY

(α/p) +∑

αTMY

(α/p) =p− 1

2− p− 1

2= 0.

192 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof 2. Na upologisje� to ginìmeno p−1∏

α=1(α/p). S� autì to ginìmeno up�rqoun

p− 1

2sÔmbola twn Legendre �sa pro 1 kai p− 1

2sÔmbola tou Legendre�sa pro −1. Opìte

p−1∏

α=1

(α/p) = (−1)p−1/2.3. Sthn Parat rhsh 2 met� to Je¸rhma 2.3.12 e�qame anaferje� stou pr¸-tou p th morf 4n+ 3 gia tou opo�ou e�qe prokÔyei ìti isqÔei(p− 1

2

)

! ≡ ±1mod p.Efarmìzonta to L mma 4.2.1, t¸ra mporoÔme na kajor�soume pìte o(p− 1

2

)

! e�nai +1 kai pìte e�nai −1mod p. SÔmfwna me to Pìrisma 4.1.2,o −1 e�nai èna TMU. Sunep¸ sthn per�ptwsh pou èqoume (p− 1

2

)

! ≡

−1mod p (ant�stoiqa (p− 1

2

)

! ≡ 1mod p) autì shma�nei ìti o (p− 1

2

)

!e�nai TMU (ant�stoiqa o (p− 1

2

)

! e�nai TU). T¸ra, èqoume((

p− 1

2

)

!/p

)

= (1/p)(2/p) · · ·(

p− 1

2/p

)

= (−1)κìpou κ e�nai to pl jo twn TMU metaxÔ twn arijm¸n 1, 2, 3 . . . ,p− 1

2.Sunep¸ an o κ e�nai �rtio (ant�stoiqa perittì ) tìte o (p− 1

2

)

! e�naiTU (ant�stoiqa TMU) kai �ra prèpei (p− 1

2

)

! ≡ 1mod p (ant�stoiqa(p− 1

2

)

≡ −1mod p).4. Na upologisje� to (−520/3). E�nai 520 = 23 · 5 · 13. Opìte(−520/3) = (−1)3−1/2(2/3)3(5/3)(13/3).All�

5 ≡ 2mod 3, 13 ≡ 1mod 3 kai (2/3) = −1.

4.2. To SÔmbolo tou Legendre 193Sunep¸ (−520/3) = (−1)(−1)3(2/3)(1/3) = −1,kaj¸ , p�nta èqoume (1/p) = 1 gia k�je perittì pr¸to p kai an α ≡

βmod p, tìte (α/p) = (β/p).Parat rhsh. An ant� gia 3 sto prohgoÔmeno par�deigma e�qame èna pr¸top arket� meg�lo, tìte ja tan toul�qiston qronobìro kai adÔnaton na upo-log�zame me ton �dio trìpo an to 2 e�nai TU e�nai TMU. Up�rqei ìmw �llo trìpo na upolog�zoume to (2/p) kai autì ofe�letai se mia èxupnh idèa touGauss. Aut èqei w ex : Upenjum�zoume ìti sthn apìdeixh tou Jewr mato tou Euler 2.3.12, gia m = p (pou antistoiqe� to mikrì Je¸rhma tou Fermat) tokr�shmo shme�o tan ìti gia ènan akèraio α me (α, p) = 1 pa�rnoume thn isotim�aαp−1(1 · 2 · · · (p − 1)) ≡ 1 · 2 · 3 · · · (p − 1)mod p kai katìpin apale�foume tonarijmì 1 · 2 · · · (p− 1). Opìte gia α = 2, èqoume thn isotim�a(∗) 2p−1(1 · 2 · · · (p − 1)) ≡ 1 · 2 · · · (p− 1)mod pkai sunep¸ 2p−1 ≡ 1mod p.Epeid apì to Krit rio tou Euler èqoume

2p−1/2 ≡ (2/p)mod pja prèpei h isotim�a (∗) na diamorfwje� kat�llhla oÔtw ¸ste ant� tou pa-r�gonta 2p−1 na èqoume ton 2p−1/2 kai oi arijmo� 1, 2, . . . , p − 1 na antikata-stajoÔn me �llou sugkekrimènou arijmoÔ . Autì epitugq�netai eis�gonta thn ènnoia tou “misoÔ” sust mato mod p pou or�sjhke apì ton Gauss. Au-tì e�nai èna opoiod pote sÔnolo A = {α1, α2, . . . , ακ} ìpou κ =p− 1

2kaioi akèraioi αi e�nai antiprìswpoi antistrèyimwn (an� dÔo mh isot�mwn) kl�-sewn mod p me thn ex idiìthta: An α e�nai èna opoiosd pote akèraio tì-te h α ≡ αi mod p α ≡ −αi mod p gia k�poio i, i = 1, 2, . . . , κ. Me�lla lìgia to A e�nai misì sÔsthma mod p an to sÔnolo A ∪ −A e�nai ènapl re sÔsthma antipros¸pwn pr¸twn kl�sewn upolo�pwn mod p. Gia par�-

194 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof deigma, to sÔnolo A ={

1, 2, . . . ,p− 1

2

} e�nai èna tètoio sÔsthma, afoÔ toA ∪ −A =

{

− p− 1

2, . . . ,−1, 1, . . . ,

p− 1

2

} e�nai pl re .A jewr soume t¸ra ton pr¸to arijmì p = 11 kai A = {1, 2, 3, 4, 5}. Efar-mìzonta thn �dia mèjodo ìpw sthn apìdeixh tou Jewr mato tou Euler giaα = 2, pa�rnoume

2 · 1 ≡ 2mod 11

2 · 2 ≡ 4mod 11

2 · 3 ≡ −5mod 11

2 · 4 ≡ −3mod 11

2 · 5 ≡ −1mod 11

⇒ 25(1 · 2 · 3 · 4 · 5) ≡ (−1)3(1 · 2 · 3 · 4 · 5)mod 11.

Dhlad , 25 ≡ −1mod 11, afoÔ (5!, 11) = 1. Opìte apì to Krit rio tou Eulerto 2 e�nai èna TMU mod11. To �dio mporoÔme na efarmìsoume gia opoiod poteα. Fereipe�n, an α = 5 èqoume

5 · 1 ≡ 5mod 11

5 · 2 ≡ −1mod 11

5 · 3 ≡ 4mod 11

5 · 4 ≡ −2mod 11

5 · 5 ≡ 3mod 11

⇒ 55(1 · 2 · 3 · 4 · 5) ≡ (−1)2(1 · 2 · 3 · 4 · 5)mod 11.

Opìte 55 ≡ 1mod 11 kai �ra to 5 e�nai èna TU mod 11.Autì o trìpo kajorismoÔ twn TU mod 11 mpore� na jewrhje� w ènaènausma gia to ex genikì apotèlesma.4.2.3 L mma (To L mma tou Gauss). 'Estw p = 2κ+ 1 èna perittì pr¸to arijmì kai A = {α1, . . . , ακ} èna misì sÔsthma mod p. An α e�nai èna akèraio me (α, p) = 1, gr�foumeααi ≡ (−1)s(i)αt(i) mod p

4.2. To SÔmbolo tou Legendre 195gia k�je αi ∈ A, ìpou s(i) ∈ {0, 1} kai t(i) ∈ {1, 2, . . . , κ}. Tìteακ ≡

κ∏

i=1

(−1)s(i) mod p.Opìte, epeid κ =p− 1

2, apì to Krit rio tou Euler, o α e�nai TU TMU anant�stoiqa to pl jo twn s(i) pou isoÔtai me 1 e�nai �rtio perittì. Dhlad

(α/p) = (−1)

κPi=1

s(i)=

1 an κ∑

i=1s(i) ≡ 0mod 2

−1 an κ∑

i=1s(i) ≡ 1mod 2Apìdeixh. Pollaplasi�zoume ìle ti isotim�e ααi ≡ (−1)s(i)αt(i) mod p gia

i ∈ {1, 2, . . . , κ} kai èqoumeακ

κ∏

i=1

αi ≡κ

∏

i=1

(−1)s(i)αt(i) mod p.'Estw i 6= j, i, j ∈ {1, . . . , κ}, tìte αt(i) 6= αt(j). Pr�gmati, an tan αt(i) = αt(j),tìte apì ti dÔo isotim�e ααi ≡ (−1)s(i)αt(i) mod p kai ααj ≡ (−1)s(j)αt(j) mod pja pa�rname thn isotim�a

(ααi)(ααj)∗ ≡ αiα

∗j ≡ (−1)s(i)−s(j)αt(i)α

∗t(j) mod p ≡ (−1)s(i)−s(j) mod p,ìpou me ta aster�kia sumbol�zoume ti ant�strofe kl�sei mod p. Dhlad jae�qame αi ≡ αj mod p αi ≡ −αj mod p. All� epeid αi, αj ∈ A kai i 6= j,autì den mpore� na sumbe�. Sunep¸ prèpei t(i) 6= t(j), dhlad αt(i) 6= αt(j).Autì shma�nei ìti kaj¸ to αi diatrèqei ìlou tou akèraiou tou sunìlou A,to αt(i) kai autì diatrèqei ìlou tou akèraiou tou A. 'Ara gia to ginìmenìtou èqoume

κ∏

i=1

αi =

κ∏

i=1

αt(i).

196 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof Epeid autì to ginìmeno de diaire�tai dia tou p, mporoÔme na to apale�youme apìthn isotim�aακ

κ∏

i=1

αi ≡κ

∏

i=1

(−1)s(i)κ

∏

i=1

αt(i) mod pgia na p�roume telik� th sqèsh pou zht�me.4.2.4 Pìrisma.(2/p) = (−1)p

2−1/8 =

{

1 an p ≡ 1 − 1mod 8

−1 an p ≡ 3 5mod 8.Apìdeixh. Upojètoume ìti p = 4n + 1. Dhlad κ =p− 1

2= 2n. jewroÔme tomisì sÔsthma A =

{

1, 2, . . . ,p− 1

2

} kai èqoume2 · 1 ≡ 2mod p

2 · 2 ≡ 4mod p...2 · n ≡ 2nmod p

2 · (n+ 1) ≡ 2n+ 2 ≡ (−2n + 1)mod p ( afoÔ 2n + 1 ≡ −2nmod p)

2 · (n+ 1) ≡ 2n+ 4 ≡ (−2n + 3)mod p...2 · (2) ≡ −1mod p.SÔmfwna me to sumbolismì pou uiojet same sto L mma tou Gauss, jètonta

α1 = 1, α2 = 2, . . . , ακ = 2n =p− 1

2, ed¸ blèpoume ìti gia ta pr¸ta i =

1, 2, . . . , n =p− 1

4èqoume s(i) = 0, en¸ gia ta teleuta�a i = n + 1, n +

2, . . . , 2n =p− 1

2èqoume s(i) = 1. 'Ara 2n

∑

i=1s(i) = n, opìte

(2/p) = (−1)n.

4.2. To SÔmbolo tou Legendre 197All� o n e�nai �rtio an kai mìnon an o p2 − 1

8e�nai �rtio , afoÔ p2 − 1

8=

1

8(p − 1)(p + 1) =

1

84n(4n + 2) = n(2n + 1). S� aut thn per�ptwsh pr�gmatièqoume

(2/p) = (−1)p2

−18 .Upojètoume t¸ra ìti p = 4n − 1, opìte 2(n − 1) =

p− 1

2− 1 kai 2n ≡

−p− 1

2mod p. Autì shma�nei ìti 2 · αi ∈ A gia i = 1, 2, . . . , n − 1 =

p− 3

4,en¸ gia i = n, n + 1, . . . , 2n − 1 =

p− 1

2, èqoume 2 · αi 6∈ A (dhlad to 2αie�nai isìtimo me k�poion akèraio sto −A). Ed¸ p�li èqoume jèsei α1 = 1, α2 =

2, . . . , ακ = 2n− 1 =p− 1

2kai sunep¸ p�li èqoume2n−1∑

i=1

s(i) =2n−1∑

i=n

1 = n.'Ara(2/p) = (−1)n.All� t¸ra 1

8(p2−1) =

1

8(p−1)(p+1) = n(2n−1) kai sunep¸ o n e�nai �rtio an kai mìnon an o 1

8(p2 − 1) e�nai �rtio . Dhlad , kai s� aut thn per�ptwshisqÔei

(2/p) = (−1)p2

−18 .Tèlo parathroÔme ìti kaj¸ o p e�nai perittì ja e�nai isìtimo me 1, 3, 5 7mod 8, dhlad ja e�nai th morf 8λ±1 8λ±3. An e�nai th morf 8λ±1tìte o p2 − 1/8 e�nai �rtio en¸ e�nai perittì an e�nai th morf 8λ± 3.Parathr sei .1. MporoÔme na doÔme apì mia �llh skopi� th diadikas�a pou akolouj samesthn apìdeixh tou 4.2.4. Ant� na pollaplasi�soume ep� 2 tou arijmoÔ

1, 2, . . . ,p− 1

2na jewr soume tou �rtiou arijmoÔ 2, 4, 6, . . . , p− 1 stopl re sÔsthma {1, 2, . . . , p − 1}, na tou pollaplasi�soume kai katìpinna ex�goume apì ton kajèna ton par�gonta 2, opìte to ginìmeno autì

198 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof ja gr�fetai w 2p−12

(p− 1

2

)

!. Katìpin na jewr soume tou arijmoÔ sto pl re sÔsthma {

− p− 1

2, . . . ,−1, 1, . . . ,

p− 1

2

} pou e�nai isìtimoime tou 2, 4, 6, . . . , p − 1 kai na tou pollaplasi�soume. To ginìmenopou ja prokÔyei ja e�nai isìtimo mod p me 2p−12

(p− 1

2

)

!. Ep�sh autìja e�nai �so me (−1)s(p − 1/2)!, ìpou s e�nai to pl jo twn arnhtik¸narijm¸n tou sunìlou {

− p− 1

2, . . . ,−1

} pou e�nai isìtimoi mod p metou arijmoÔ 2, 4, 6, . . . , p− 1 oi opo�oi den e�nai isìtimoi me arijmoÔ tousunìlou {

1, 2, . . . ,p− 1

2

}. Dhlad to s e�nai to pl jo twn arijm¸n2, 4, 6, . . . , p− 1 pou e�nai megalÔteroi tou p− 1

2. 'Etsi ja èqoume 2

p−12 ≡

(−1)s mod p.2. Mia �llh apìdeixh tou 4.2.4 e�nai h ex . JewroÔme ti p− 1

2isotim�e

p− 1 ≡ 1(−1)1 mod p

2 ≡ 2(−1)2 mod p

p− 3 ≡ 3(−1)3 mod p

4 ≡ 4(−1)4 mod p...m ≡ p− 1

2(−1)

p−12 mod pìpou

m =

p− 1

2an o p− 1

2e�nai �rtio

p− p− 1

2an o p− 1

2e�nai perittì .Parathr¸nta ìti k�je �rtio arijmì 2, 4, 6, . . . , p − 1 emfan�zetai k�-pou sto aristerì mèlo twn isotimi¸n, pollaplasi�zonta ìle autè ti isotim�e pa�rnoume 2 · 4 · 6 · · · (p− 1) ≡

(p− 1

2

)

!(−1)1+2+···+ p−12 mod p

2p−12

(p− 1

2

)

! ≡(p− 1

2

)

!(−1)p2

−18 mod p.

4.3. O Nìmo th Antistrof 1994.3 O Nìmo th Antistrof O tetragwnikì nìmo antistrof , me ton opo�o ja asqolhjoÔme s� aut thnpar�grafo, apotele� èna apì ta wraiìtera kai shmantikìtera apotelèsmata th Jewr�a Arijm¸n. Autì susqet�zei, gia dÔo diaforetikoÔ perittoÔ pr¸tou arijmoÔ p kai q, ti lÔsei th ex�swsh x2 ≡ qmod pme ti lÔsei th ex�swsh x2 ≡ pmod qw ex : an èna apì tou dÔo pr¸tou e�nai th morf 4n+ 1 tìte kai oi dÔoexis¸sei e�nai epilÔsime kai oi dÔo den e�nai epilÔsime , en¸ an kai oi dÔopr¸toi e�nai th morf 4n+3, tìte h mia ex�swsh e�nai epilÔsimh kai h �llh dene�nai epilÔsimh. Gia par�deigma, an p = 5, ta TU mod5 e�nai to 1 kai 4 mod5.Sunep¸ gia ènan perittì pr¸to arijmì q 6= 5, x2 ≡ qmod5 èqei lÔsh an

q ≡ 1mod 5 q ≡ 4mod 5. SÔmfwna me to nìmo antistrof , an q ≡ 1mod 5 q ≡ 4mod 5 tìte h x2 ≡ 5mod q èqei lÔsh (dhlad to 5 e�nai TU mod q)anex�rthta an to q e�nai th morf 4κ+ 1 4κ+ 3. En¸ an p = 7, tìte ta TUmod 7 e�nai to 1 kai to 2 kai to 4 mod 7. O nìmo antistrof ma bebai¸neiìti an q 6= 7 e�nai èna pr¸to th morf 4n+ 3 kai h x2 ≡ qmod7 èqei lÔsh,dhlad q ≡ 1mod 7 q ≡ 2mod 7 q ≡ 4mod 7 tìte h ex�swsh x2 ≡ 7mod qden èqei lÔsh, dhlad to 7 e�nai TMU mod q.S mera o nìmo th antistrof sunhj�zetai na anafèretai w o “nìmo an-tistrof tou Gauss”. All� o pr¸to pou diatÔpwse thn eikas�a ìti o nìmo autì isqÔei tan o Euler to 1783 kai katìpin to 1785 o Legendre isqur�sjh-ke ìti ton apèdeixe. H apìdeix tou ìmw tan l�jo kai to 1798 èdwse �llhapìdeixh pou kai aut den tan pl rh kaj¸ upèjese ìti up�rqoun �peiroi topl jo pr¸toi th morf ακ+β, κ ∈ N gia dosmènou α, β ∈ Z me (α, β) = 1.Autì ìmw apede�qjei (met� ton Legendre) to 1837 apì ton Dirichlet qrhsi-mopoi¸nta th jewr�a migadik¸n sunart sewn. H pr¸th swst apìdeixh tou

200 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof tetragwnikoÔ nìmou antistrof ed¸jei to 1795 apì ton Gauss pou dhmosieÔ-jhke sto gnwstì bibl�o tou Disquisitiones Arithmeticae. O Gauss diatÔpwsekai apèdeixe autì ton nìmo qwr� na gnwr�zei th sqetik doulei� twn Euler kaiLegendre. A shmeiwje� ìti o Gauss tan tìte mìli 18 et¸n kai aisjanìtantìso uper fano gi� autì to apotèlesma pou o �dio to apèdide ston eautì touapokal¸nta to w “o nìmo antistrof tou Gauss”. 'Ew s mera èqoun dw-je� perissìtere apì 200 diaforetikè apode�xei gi� autìn to nìmo pou oi 8 ap�autè an koun ston Gauss. 'Ep�sh o Gauss maz� me ton majht tou Ferdinand

Eisenstein (pou pèjane mìli 29 et¸n) gen�keusan ton tetragwnikì nìmo anti-strof sto kubikì kai dutetragwnikì nìmo antistrof eis�gonta ta gnwst�s mera “ajro�smata tou Gauss”. To 1920 o Emil Artin diatÔpwse to “Je¸rhmaAntistrof ” pou perilamb�nei ìlou tou gnwstoÔ nìmou antistrof w eidikè peript¸sei . H perigraf autoÔ tou jewr mato apaite� arketè gn¸sei apì th sÔgqronh �lgebra kai thn Algebrik Jewr�a Arijm¸n pou e�nai pèrantou skopoÔ aut¸n twn shmei¸sewn.Mia apì ti pio aplè apode�xei tou tetragwnikoÔ nìmou antistrof pouja d¸soume amèsw pio k�tw e�nai mia parallag th tr�th apìdeixh tou Gaussh opo�a sthr�zetai sto L mma 4.2.3 tou Gauss kai mia parat rhsh tou Einse-

stein. (Mia �llh apl apìdeixh ofe�letai ston Frobenius, blèpe Shanks).To L mma tou Gauss mporoÔme na to anadiatup¸soume (w pìrism� tou)sthn ex morf .4.3.1 L mma (2h diatÔpwsh tou L mmato tou Gauss). 'Estw α èna perittì akèraio kai p èna perittì pr¸to arijmì pou de diaire� ton α. Tìte isqÔei(α/p) = (−1)

p−1/2Pi=1

hiαp

iìpou me [x] sumbol�zoume to akèraio mèro enì pragmatikoÔ arijmoÔ x, dhlad o [x] e�nai o megalÔtero akèraio pou e�nai mikrìtero apì ton x (gia par�deigma,[x] = x an kai mìnon an x ∈ Z, [−

√2] = −2, [

√2] = 1, [1

5

]

= 0, [

− 6

5

]

0 − 2).Apìdeixh. JewroÔme to misì sÔsthma A ={

1, 2, . . . ,p− 1

2

}, opìte sto 4.2.3

4.3. O Nìmo th Antistrof 201èqoume αi = i, i = 1, . . . ,p− 1

2kai αi ≡ (−1)s(i)αt(i) mod p, t(i) ∈ A. 'Eqoumedei de ìti αt(i) 6= αt(j) gia i 6= j, i, j ∈ A. 'Estw j1, j2, . . . , jr ìla ta i ∈ A giata opo�a s(i) = 1 kai jr+1, . . . , jp−12

ìla ta i ∈ A gia ta opo�a s(i) = 0. Dhlad αj

λ≡

{

−αt(jλ) mod p, λ = 1, . . . , r

αt(jλ) mod p, λ = r + 1, . . . , p−1

2 .Sunep¸ oi akèraioi αt(j1), . . . , αt(jr), αt(jr+1), . . . , αt

�j p−1

2

� e�nai mia anadi�taxh(met�jesh) twn akèraiwn 1, 2, . . . ,p− 1

2. DiairoÔme t¸ra ton αi dia tou p, giak�je i ∈ A kai èqoume apì thn Eukle�deia dia�resh

αi = pπi + ri, 0 ≤ ri < p, i ∈ A.Opìte πi =[αi

p

], afoÔ 0 ≤ rip< 1, i ∈ A. E�nai de ri 6= p

2, i ∈ A, afoÔ o p

2den e�nai akèraio .T¸ra parathroÔme ìti an ri > p

2, tìte o ri e�nai �so me ènan apì tou

p − αt(jλ), λ = 1, 2, . . . , r kai an ri < p

2tìte o ri e�nai �so me ènan apì tou

αt(jλ), λ = r + 1, . . . ,

p− 1

2. Prosjètonta ìle ti isìthte αi = p

[αi

p

]

+ ri, i ∈ Apa�rnoumeαp2 − 1

8= α

p−12

∑

i=1

i = α

p−12

∑

λ=1

jλ

= p

p−12

∑

i=1

[αi

p

]

+

p−12

∑

i=1

ri

= p

p−12

∑

λ=1

[αjλ

p

]

+r

∑

λ=1

(p− αt(jλ)) +

p−12

∑

λ=r+1

αt(jλ).Ep�sh èqoume 1 + 2 + · · · + p− 1

2=p2 − 1

8=

p−12

∑

λ=1

αt(jλ).

202 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof Afair¸nta thn teleuta�a isìthta apì thn prohgoÔmenh lamb�noume thnisìthta(α− 1)

p2 − 1

8= p

p−12

∑

λ=1

[αjλ

p

]

+ r

− 2r

∑

λ=1

αt(jλ)

= p

p−12

∑

i=1

[αi

p

]

+ r

− 2

r∑

λ=1

αt(jλ)afoÔ {

1, 2, . . . ,p− 1

2

}

={

j1, j2, . . . , jp−12

}.Epeid upojèsame ìti o α kai o p e�nai peritto� arijmo�, dhlad α ≡ p ≡1mod 2, èqoume thn isotim�a

0 · p2 − 1

8= 0 ≡ 1 ·

p−12

∑

i=1

[αi

p

]

+ r

mod 2.'Arap−12

∑

i=1

[αi

p

]

≡ −r ≡ rmod2.E�nai ìmw r =

p−12

∑

i=1s(i) kai ètsi to L mma 4.2.3 pa�rnei th morf α

p−12 ≡ (−1)

p−12P

i=1

hαip

i=

(α

p

)

mod p.

Parat rhsh. Sto prohgoÔmeno l mma mporoÔme na sumperilamb�noume kaithn per�ptwsh α = 2, diìti s� aut thn per�ptwsh èqoume [2i

p

]

= 0, afoÔ2i < p, gia i = 1, 2 . . . ,

p− 1

2. Opìte p−1

2∑

i=1

[2i

p

]

= 0 kai sunep¸ (2−1)p2 − 1

8=

p2 − 1

8≡ rmod2. 'Ara 2

p−12 = (−1)

p2−18 mod p.

4.3. O Nìmo th Antistrof 203ApodeiknÔoume t¸ra to nìmo antistrof o opo�o diatup¸netai me ta sÔm-bola tou Legendre w ex .4.3.2 Je¸rhma (Nìmo Antistrof ). 'Estw p kai q dÔo peritto� pr¸toiarijmo� di�foroi metaxÔ tou . Tìte isqÔei(q

p

)

= (−1)p−12

q−12

(p

q

) isodÔnama(q

p

)

=

(

p

q

) an p ≡ 1mod 4 q ≡ 1mod 4

−(

p

q

) an p ≡ q ≡ 3mod 4.Apìdeixh. 'Estw Q =

p−12

∑

i=1

[qi

p

] kai P =

q−12

∑

j=1

[pi

q

]. Tìte apì to L mma touGauss 4.3.1 èqoume

(q

p

)(p

q

)

= (−1)Q+P .ApodeiknÔoume ìti Q+P = p−12

q−12 . Autì prokÔptei apì thn ex parat rhshpou èkane o Eisenstein sto Gauss. O arijmì Q + P e�nai akrib¸ to pl jo twn akèraiwn shme�wn entì tou orjog¸niou parallhlìgrammou sto ep�pedo mekorufè ta shme�a (0, 0),

(p

2, 0

)

,(p

2,q

2

) kai (

0,q

2

). Dhlad anS =

{

(x, y) ∈ Z × Z/1 ≤ x ≤ p− 1

2, 1 ≤ y ≤ q − 1

2

}tìte Q + P = |S|. Pr�gmati, èstw E h euje�a pou dièrqetai apì ti korufè (0, 0) kai (p

2,q

2

). H ex�swsh aut th euje�a e�nai py = qx, dhlad y =q

px.Epeid to kl�sma q

pe�nai an�gwgo (afoÔ (p, q) = 1), kanèna shme�o tou S debr�sketai p�nw sthn euje�a E, diìti an (x0, y0) ∈ S me y0 =

q

px0 tìte to x0 jaèprepe na e�nai pollapl�sio tou p, all� 0 < x0 < p.T¸ra ta shme�a tou S pou br�skontai k�tw apì thn euje�a E e�nai eke�na ta

(x, y) ∈ S me py < qx.

204 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof Sunep¸ ta shme�a pou br�skontai k�tw apì thn euje�a E, an koun sto Skai èqoun stajer tetmhmènh x = i, e�nai ta shme�a (i, 1), (i, 2), . . . ,

(

i,[qi

p

]

)pou to pl jo tou e�nai [qi

p

]. Sunep¸ to pl jo twn shme�wn tou S poubr�skontai k�ta apì thn euje�a E e�nai �so meQ =

p−12

∑

i=1

[qi

p

]

.Me ton �dio trìpo, ta shme�a p�nw apì thn euje�a E kai an koun sto S e�naieke�na ta (x, y) ∈ S me py > qx. Sunep¸ aut� ta shme�a me stajer tetagmènhy = j, e�nai ta shme�a (1, j), (2, j), . . . ,

(

[pj

q

]

, j

). 'Ara to pl jo twn shme�wntou S pou br�skontai p�nw apì thn euje�a E e�nai �so meP =

q−12

∑

j=1

[pj

q

]

.Telik� to pl jo |S| =p− 1

2

q − 1

2twn shme�wn tou S e�nai �so me Q + P .(Shmei¸noume ìti sthn prohgoÔmenh apìdeixh de qrhsimopoi same ìti oi p kai qe�nai pr¸toi all� ìti e�nai sqetik� pr¸toi metaxÔ tou peritto� arijmo�. Dhlad

Q+P =p− 1

2

q − 1

2isqÔei gia opoiosd pote perittoÔ sqetik� pr¸tou metaxÔtou arijmoÔ p kai q).Sun jw gia to sÔmbolo tou Legendre (α/p), o akèraio α lègetai “arijmht ”kai o pr¸to arijmì p lègetai “paronomast ”. Gnwr�zoume ìti an oi “arijmhtè ”

α kai α′ twn (α/p) kai (α′/p) e�nai isìtimoi mod p tìte(α

p

)

=(α′

p

)

.Gia dÔo diaforetikoÔ “paronomastè ” p kai p′, apì to nìmo antistrof pro-kÔptei to ex pìrisma.4.3.3 Pìrisma. 'Estw p, p′ kai q trei diaforetiko� peritto� pr¸toi arijmo�.An p′ ≡ ±pmod4q tote(q

p

)

=( q

p′

)

.

4.3. O Nìmo th Antistrof 205Dhlad to sÔmbolo tou Legendre(q

p

) exart�tai mìno apì thn kl�sh upolo�pwnpmod4q. Epiplèon autì pa�rnei thn �dia tim me paranomastè pr¸tou arijmoÔ pou e�nai isìtimoi mod4q e�te me to p e�te me to −p.Apìdeixh. 'Estw q ≡ 1mod 4, dhlad o q − 1

2e�nai �rtio . Tìte

(p′

q

)( q

p′

)

= 1 =(p

q

)(q

p

)

.An p′ ≡ pmod q, tìte (p′

q

)

=(p

q

) opìte ( q

p′

)

=(q

p

). Dhlad an q ≡ 1mod 4kai p′ ≡ pmod q tìte to (q

p

) exart�tai mìno apì thn kl�sh pmod q. Anp′ ≡ −pmod q, tìte (p′

q

)

=(−pq

)

=(−1

q

)(p

q

)

=(p

q

) afoÔ (−1

q

)

=

(−1)q−12 = 1. Dhlad kai s� aut thn per�ptwsh to (q

p

) exart�tai mìno apìthn kl�sh pmod q.'Estw t¸ra q ≡ −1mod 4, dhlad o q − 1

2e�nai perittì . An p′ ≡ pmod 4qtìte p′ ≡ pmod4 kai p′ ≡ pmod q. Opìte p′ − 1

2≡ p− 1

2mod2 kai (p′

q

)

=(p

q

). Apì to nìmo antistrof pa�rnoume(p′

q

)( q

p′

)

= (−1)p′−1

2q−12 = (−1)

p−12

q−12 =

(p

q

)(q

p

)

.'Ara ( q

p′

)

=(q

p

).Ep�sh an p′ ≡ −pmod4q tìte p′ ≡ −pmod4 kai p′ ≡ −pmod q. Opìtep′ − 1

2=

p+ 1

2mod 2 kai (p′

q

)

=(−pq

)

= (−1)q−12

(p

q

)

= −(p

q

). Apì tonìmo antistrof pa�rnoume(p′

q

)( q

p′

)

= (−1)p′−1

2 = (−1)p+12 = (−1)

p−12

+1 = −(−1)p−12 = −

(p

q

)(q

p

)

.'Ara p�li èqoume ( q

p′

)

=(q

p

).Lamb�nonta upìyh to Pìrisma 4.2.2, kai 4.2.4, tìte to 4.3.3 mpore� naanadiatupwje� sthn ex genik morf .

206 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof 4.3.4 Pìrisma. 'Estw α > 1 èna fusikì kai p 6= q dÔo pr¸toi peritto�arijmo� pou de diairoÔn ton α. An p ≡ ±qmod4α, tìte(α

p

)

=(α

q

)

.Sthn arq tou kefala�ou autoÔ xekin same me to prìblhma kajorismoÔ twnTU mod p, gia ènan perittì pr¸to arijmì p kai e�dame ìti to Krit rio tou Eulerkai to L mma tou Gauss d�noun sqetikè apant sei s� autì to prìblhma. Tojemelei¸de ìmw er¸thma sth jewr�a twn tetragwnik¸n upolo�pwn sun�stataisthn “anastrof ” autoÔ tou probl mato , dhlad sto kajorismì ìlwn twnperitt¸n arijm¸n p gia tou opo�ou èna fusikì arijmì α > 1 e�nai TUmod p. H ap�nthsh s� autì to prìblhma d�detai apì to nìmo antistrof mèsw tou Por�smato 4.3.4 to opo�o ousiastik� anafèrei ìti “oi pr¸toi peritto�diairète p, arijm¸n th morf n2−q, e�nai th morf 4qκ+β2 4qκ−β2, giak�poio perittì arijmì β, ìpou q e�nai èna dosmèno perittì pr¸to arijmì .Parade�gmata. A sunoy�soume katarq� ti basikè idiìthte tou sumbìloutou Legendre.1. (1

p

)

= 1 gia k�je pr¸to perittì p.2. (−1

p

)

= (−1)p−12 .3. (2

p

)

= (−1)p2

−18 .4. An α ≡ βmod p, tìte (α

p

)

=(β

p

).Idia�tera isqÔei: (α+ κp

p

)

=(α

p

), κ ∈ Z.5. (αβ

p

)

=(α

p

)(β

p

).6. (αβ2

p

)

=(α

p

).7. (p

q

)

= (−1)p−12

q−12

(q

p

)ìpou α, β e�nai pr¸toi pro ton p kai q, p 6= q pr¸toi peritto� arijmo�. Shmei¸-noume ìti h idiìthta 4) maz� me ton nìmo antistrof (idiìthta 7)) e�nai polÔqr simh gia ton upologismì tou sumbìlou (α

p

) kaj¸ an α = ±2tpt1 · · · ptss

4.3. O Nìmo th Antistrof 207e�nai h an�lush tou α se perittoÔ pr¸tou p1, . . . , ps tìte(α

p

)

=(±1

p

)(2

p

)t(p1

p

)t1 · · ·(ps

p

)ts.MporoÔme na upojèsoume ìti ìloi oi pr¸toi pi e�nai mikrìteroi tou p, diìti an giak�poio i, pi > p, tìte diair¸nta èqoume (pi

p

)

=(κp+ r

p

)

=(r

p

) me 0 < r < p.To (r

p

) analÔetai se ginìmeno sumbìlwn (rip

) ìpou ri e�nai oi pr¸toi diairète tou r kai ri < p. T¸ra gia ta sÔmbola (pi

p

), pi < p, efarmìzoume thn idiìthta7, dhlad ta antikajistoÔme me (−1)

�pi−1

2

�( p−1

2 )( p

pi

). DiairoÔme to p dia pi kaisuneq�zoume èw ìtou fj�soume se sÔmbola th morf (±1

q

), gia k�poiou pr¸tou q.A efarmìsoume thn prohgoÔmenh mèjodo sto sÔmbolo ( 11

1729

) upojètonta ìti o 1729 e�nai pr¸to arijmì . 'Eqoume( 11

1729

)

= (−1)5.864(1729

11

)

=(157 · 11 + 2

11

)

=( 2

11

)

= −1kaj¸ 11 ≡ 3mod 8 (blèpe 4.2.4).All� ep�sh èqoume ìti 111729−1

2 ≡( 11

1729

)

mod 1729. (Krit rio Euler).Dhlad 11864 ≡ −1mod1729.Upolog�zonta ìmw to 11864 mod 1729 br�skoume ìti 11864 ≡ 1mod 1729. (E�-nai 1112 ≡ 1mod 1729 kai 864 = 12·72). Sunep¸ h upìjes ma ìti o 1729 e�naipr¸to tan l�jo . 'Ara o 1729 den e�nai pr¸to arijmì . Autì to par�deigmaupodeiknÔei pw mporoÔme na qrhsimopoioÔme gia na elègqoume an èna arijmì e�nai pr¸to . Aut h mèjodo e�nai qr simh ìtan e�nai eÔkolo na thn efarmì-zoume se meg�lou arijmoÔ , giat� ed¸ eÔkola prokÔptei ìti 1729 = 7 · 13 · 17.Ton arijmì 1729 sun jw to jumìmaste apì to ex “par�doxo gegonì ”pou perigr�fei o G.H. Hardy sto bibl�o tou: “Mia Apolog�a enì Majhma-tikoÔ”. 'Otan episkèfjhke o Hardy ton Ramanujan, ìpou noshleuìtan, senosokome�o sthn Aggl�a, o Ramanujan r¸thse ton Hardy an o arijmì tou

208 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof tax� pou ton e�qe metafèrei tan k�poio endiafèron arijmì . O Hardy ap�nth-se pw den nom�zei ìti tan endiafèfon, apl� tan o arijmì 1729. Amèsw oRamanujan ap�nthse: Antijètw , autì o arijmì e�nai shmantikì epeid e�naio mikrìtero fusikì arijmì pou mpore� na grafte� w �jroisma dÔo kÔbwn medÔo diaforetikoÔ trìpou . Pr�gmati,

1729 = 103 + 93 = 123 + 13.To Krit rio tou Euler den e�nai eÔkolo na efarmìzetai gia na elègqoumean èna meg�lo arijmì e�nai TU modulo ènan meg�lo pr¸to p. S� autè ti peript¸sei upolog�zoume to sÔmbolo tou Legendre.Gia par�deigma, den e�nai eÔkolo na upolog�soume thn kl�sh 37691−1

2 mod 629.All� ant� autoÔ upolog�zoume to o sÔmbolo tou Legendre:( 37

621

)

= (−1)37−1

2691−1

2

(691

37

)

=(691

37

)

=(18 · 37 + 25

37

)

=( 5

37

)2= 1.'Ara to 37 e�nai TU mod 691. Dhlad o upologismì tou sumbìlou tou Legen-

dre, qrhsimopoi¸nta ti idiìthte 1)-7), ma lèei ìti ex�swsh x2 ≡ 37mod 691èqei lÔsh, all� ìmw den ma d�nei mèjodo gia na broÔme aut th lÔsh. Ep�sh gia na upolog�soume to sÔmbolo tou Legendre pro�pìjesh e�nai na gnwr�zoumean o “paronomast ” e�nai pr¸to arijmì .A upolog�soume to (299

359

) me thn pro�pìjesh ìti to 359 e�nai pr¸to .(299

359

)

=( 13

359

)( 23

359

)

( 13

359

)

=(359

13

)

=(13 · 27 + 8

13

)

=( 2

13

)3= −1(afoÔ 13 ≡ 5mod 8) ( me to Krit rio tou Euler: ( 2

13

)

= 26 = 64 ≡ 12 ≡−1mod 13)

( 23

359

)

= −(359

23

)

= −(23 · 15 + 14

23

)

= −(14

23

)

= −( 2

23

)( 7

23

)

= −(+1)( 7

23

)

= −( 7

23

)

= −(

−(23

7

)

)

= −(

−(3 · 7 + 2

23

)

)

=( 2

23

)

= 1.

4.3. O Nìmo th Antistrof 209'Ara (299

359

)

= (−1)(+1) = −1.Dhlad , h x2 ≡ 299mod 359 den èqei lÔsh.Na luje� h ex�swsh x2 ≡ 197mod 1241. H an�lush tou 1241 se pr¸tou e�nai 1241 = 73 · 17. Opìte h dosmènh ex�swsh èqei lÔsh an kai mìnon an oidÔo exis¸sei x2 ≡ 197mod 73 kai x2 ≡ 197mod 17 èqoun lÔsh. Gia thn pr¸thèqoume(197

73

)

=(2 · 73 + 51

p

)

=(51

73

)

=(−22

73

)

=(−1

73

)( 2

73

)(11

73

)all� (−1

73

)

= 1, ( 2

73

)

= 1, (11

73

)

=(73

11

)

=( 7

11

)

= −(11

7

)

= −(4

7

)

= −1.Sunep¸ (197

73

)

= −1, dhlad h x2 ≡ 197mod 73 den èqei lÔsh kai �ra kai hx2 ≡ 197mod 1241 den èqei lÔsh. En¸ epeid

(23

41

)

=(41

23

)

=(18

23

)

=( 2

23

)( 3

23

)2= 1kai

(23

97

)

= −(97

23

)

= −( 5

23

)

= −(23

5

)

= −(3

5

)

= −(5

3

)

= −(2

3

)

= −(−1) = 1h ex�swsh x2 ≡ 23mod 41 · 97 èqei lÔsh.Sta prohgoÔmena dÔo parade�gmata efarmìsame to Je¸rhma 3.4.1 apì toopo�o oi lÔsei th ex�swsh x2 − α ≡ 0modm, gia m ∈ N, kajor�zontai apìti lÔsei twn exis¸sewn x2−α ≡ 0mod pλ, ìpou pλ e�nai h mègisth dunat enì pr¸tou p pou diaire� to m. Gia ti lÔsei th x2 − α ≡ 0mod pλ efarmìzoumeto Je¸rhma 3.4.3 (L mma Hensel). Diakr�noume dÔo peript¸sei 1h per�ptwsh: p > 2. 'Estw x0 e�nai mia lÔsh th f(x) = x2 −α ≡ 0mod pκ, me0 ≤ x0 ≤ pκ−1, tìte, an f ′(x0) = 2x0 6≡ 0mod p up�rqei mia monadik lÔsh th x2−α ≡ 0mod pκ+1 th morf x0+pκtmod pκ+1 gia k�poio t = 0, 1, . . . , p−1.Autì to t d�detai (blèpe apìdeixh tou 3.4.3) apì thn isotim�a

t ≡(

−(2x0)∗ (x2

0 − α)

pκ

)

· mod p.ìpou (2x0)∗ mod p e�nai h ant�strofh kl�sh th 2x0 mod p.

210 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof Sunep¸ an x0 e�nai mia lÔsh th x2 − α ≡ 0mod p, (opìte (α

p

)

= 1) tìtef ′(x0) = 2x0 6≡ 0mod p (èqoume upojèsei ìti α 6≡ 0mod p) afoÔ p > 2. 'Ara hx2 − α ≡ 0mod p2 èqei mia monadik lÔsh th morf x0 + pt = y0, gia k�poiot = 0, 1, . . . , p− 1. Gia ton �dio lìgo kai h x2 − α ≡ 0mod p3 èqei mia monadik lÔsh th morf y0 + p2t, gia k�poio t = 0, 1, . . . , p − 1. Suneq�zonta me ton�dio trìpo, sumpera�noume ìti h x2 − α ≡ 0mod pκ+1 èqei monadik lÔsh pouantistoiqe� sthn lÔsh x0 th x2 − α ≡ 0mod p. Sunep¸ h x2 − α ≡ 0mod pλèqei dÔo mh-isìtime lÔsei kami� lÔsh an h x2−α ≡ 0mod p èqei lÔsh (opìteèqei dÔo mh isìtime lÔsei ) den èqei kam�a lÔsh ant�stoiqa. Autì prokÔpteikai apì to Pìrisma 3.4.4 mpore� na apodeiqje� anex�rthta w ex : An hx2 − α ≡ 0mod pλ èqei mia lÔsh thn x0 tìte profan¸ èqei kai thn −x0 poue�nai mh isìtimh th x0 afoÔ 2x0 6≡ 0mod pr. An x1 e�nai mia opoiad pote lÔshth x2 − α ≡ 0mod pλ tìte (x1 − x0)(x1 + x0) ≡ x2

1 − x20 ≡ 0mod pλ. 'Ara

pλ|x1 − x0 pλ|x1 + x0 all� to pλ de diaire� kai ta dÔo, diìti diaforetik� to pja ta diairoÔse kai �ra ja diairoÔse kai th diafor� tou pou e�nai 2x0 kai autìe�nai �topo. Autì shma�nei ìti x1 ≡ x0 mod pr x1 ≡ −x0 mod pλ.Par�deigma. JewroÔme thn isotim�a x2 ≡ 5mod 113. Epeid h x2 ≡ 5mod 11èqei lÔsh thn x0 = 4 (kai thn −4), dhlad ( 5

11

)

= 1, h dosmènh isotim�a èqeilÔsh. Gia na broÔme ti dÔo lÔsei aut th isotim�a , lÔnoume pr¸ta thnx2 ≡ 5mod 112. Oi lÔsei aut e�nai ±(4 + t · 11)mod 112, ìpou

t = −(2 · 4)∗ 42 − 5

11mod11, (2 · 4)∗ = 7.'Ara t ≡ −7 ≡ 4mod 11. Opìte oi lÔsei e�nai ±(4 + 4 · 11) = ±48. T¸ra oilÔsei th x2 ≡ 5mod 113 e�nai ±(48+t·112) me t = −(2·48)∗ 482 − 5

112mod11 =

10mod 11. Opìte oi zhtoÔmene lÔsei e�nai ±1258mod 113.A broÔme t¸ra tou perittoÔ pr¸tou p gia tou opo�ou to 3, to 5, to 7,to 11 kai to 13 e�nai TU mod p. Dhlad gia poiou p èqoume(3

p

)

=(5

p

)

=(7

p

)

=(11

p

)

=(13

p

)

= 1Diakr�noume autoÔ pou e�nai th morf 4κ+ 1 kai autoÔ th morf 4κ+ 3.

4.3. O Nìmo th Antistrof 211O 5 kai o 13 e�nai th morf 4κ+1 en¸ oi upìloipoi e�nai th morf 4κ+3.Epeid (5

p

)

=(13

p

)

= 1 ja prèpei na èqoume kai (p

5

)

=( p

13

)

= 1,e�te o p e�nai th morf 4κ + 1 e�te e�nai th morf 4κ + 3. En¸ epeid (3

p

)

=(7

p

)

=(11

p

)

= 1, tìte ja prèpei (p

3

)

=(p

7

)

=( p

11

)

= 1 gia tou pth morf 4κ+1 kai (p3

)

=(p

7

)

=( p

11

)

= −1 gia tou p th morf 4κ+3.All� (p

5

)

= 1 shma�nei ìti o p e�nai TU mod5. Sunep¸ o p prèpei na e�naiisìtimo me ènan apì tou 1, 2, 3, 4 pou e�nai TU mod5. Ta TU mod5 e�naiisìtima me tou 12 = 1 22 = 4 modulo 5. 'Ara o p prèpei na e�nai isìtimo meton 1mod 5 me ton −1mod5, dhlad prèpei na e�nai èna pr¸to th morf 1 + 5κ èna pr¸to th morf −1 + 5κ. Shmei¸noume ìti oi arijmo� th morf 1+ ·5 ·κ kai −1+4 ·5 ·κ e�nai th morf 1+5κ kai −1+5κ ant�stoiqa.An jèloume na taxinom soume tou pr¸tou p s� autoÔ th morf 4κ+ 1 kais� autoÔ th morf 4κ+ 3 tìte lÔnoume ta sust mata:(a) x ≡ 1mod 5

x ≡ 1mod 5(b) x ≡ −1mod 5

x ≡ 1mod 4(g) x ≡ 1mod 5

x ≡ 3mod 4kai(d) x ≡ −1mod 5

x ≡ 3mod 4.Gia to (a) h lÔsh e�nai x ≡ 1mod 20, gia to (b) e�nai x ≡ 32 mod20, gia to(g) e�nai x ≡ −32 mod 20 kai gia to (d) e�nai x ≡ −1mod 20. Dhlad , ìloi oipr¸toi peritto� diairète (di�foroi tou 5) arijm¸n th morf n2 − 5 e�nai th morf 1 + 20κ, −1 + 20κ, 32 + 20κ kai −32 + 20κ, gia k�poia κ ∈ Z.Me ton �dio akrib¸ trìpo br�skoume ìti ìloi oi peritto� pr¸toi (di�foroitou 13) pou e�nai diairète arijm¸n th morf n2 − 13 e�nai th morf 1+52κ, −1+52κ, 32+52κ, −32+52κ, 52+52κ, −52+52κ, 72+52κ, −72+52κ,92 + 52κ, −92 + 52κ, 112 + 52κ −112 + 52κ.A exet�soume t¸ra thn per�ptwsh (7

p

)

= 1. Epeid 7 = 4 ·1+3, jewroÔmetou p th morf 4κ+ 1 gia tou opo�ou prèpei na èqoume (p

7

)

= 1 kai ìlou tou p th morf 4κ+3 gia tou opo�ou prèpei na èqoume (p

7

)

= −1. Dhlad ja prèpei na jewr soume ìlou tou p th morf 4κ + 1 pou e�nai TU mod7

212 Kef�laio 4. Tetragwnik� Upìloipa kai o Nìmo Antistrof kai ìlou tou p th morf 4κ+ 3 pou e�nai TMU mod7. Ed¸ ta TU mod7e�nai isìtima me ènan apì tou 1 = 12, 4 = 22, 2 ≡ 32 mod7. 'Ara ta TMUmod 7 e�nai isìtima me ènan apì tou 3, 5 6 mod7. 'Etsi èqoume na lÔsoumeta sust mata(a) x ≡ 1mod 7

x ≡ 1mod 4(b) x ≡ 2mod 7

x ≡ 1mod 4(g) x ≡ 4mod 7

x ≡ 1mod 7(d) x ≡ 3mod 7

x ≡ 3mod 4(e) x ≡ 5mod 7

x ≡ 3mod 4(st) x ≡ 6mod 7

x ≡ 3mod 4 .Oi lÔsei aut¸n e�nai x ≡ 1mod 28, 32 mod28, 52 mod 28, −52 mod28,−32 mod28 kai−1mod28 ant�stoiqa. Dhlad oi pr¸toi p pou zht�me na broÔmee�nai th morf ±1+28κ, ±32+28κ, ±52+28κ gia k�poia κ ∈ Z. H per�ptwshtwn pr¸twn 3 kai 11 af netai w �skhsh.

Shmei¸seic gia to m jhma JEWRIAS ARIJMWN (D. Derizi¸thc) · 1.1.5 Orismìc. Oi akèraioi … kai...

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Transcript of Shmei¸seic gia to m jhma JEWRIAS ARIJMWN (D. Derizi¸thc) · 1.1.5 Orismìc. Oi akèraioi … kai...