NÚMEROS PRIMOS:

EL RELOJ DE SERGIO ASCHERO

(UN ORDENAMIENTO LÓGICO DE LO APARENTEMENTE CAOTICO)

A Mirta Karp: el infinito de todos mis amores

INTRODUCCIÓN

Los matemáticos han intentado en vano desde hace mucho tiempo descubrir alguna secuencia en el orden de los números primos, pero tengo razones para creer que éste es un misterio en el que la mente humana jamás podrá penetrar.

Leonhard Euler

Resumen

Las matemáticas tienen sus propios problemas sin resolver.Uno de ellos tiene que ver con el ordenamiento de los números primos.Con respecto a la generación de los mismos, la Criba de Aschero produce estos números limpiándolos de los compuestos y simplificando el modelo establecido por Eratóstenes con su famosa Criba.Sergio Aschero cree haber encontrado un modelo aritmético que ordena lo aparentemente caótico, y partiendo del Reloj de Sergio Aschero se puede explicar la lógica de la aparente discontinuidad de los números primos.Y eso es lo que veremos a continuación:

1

Análisis

a) el reloj de los números primos tiene 18 horas.b) los números primos aparecen en 6 horas determinadas y su expansión

hacia el infinito se produce solamente en 6 de esas horas.c) las horas del reloj: 1, 5, 7, 11, 13, 17.

Todo número primo es congruente con 1,5,7,11,13,17 (módulo 18), ie, que todo número primo es de la forma  , con 

Sabemos que, dados   únicos tales que Esto es lo que se conoce como división entera (de   entre  ), con   cociente,   residuo.

También sabemos que, por definición,   no será primo si es producto de dos números, de modo que  no será primo si   o cualquiera de sus múltiplos debido a que 18=2·3·3, de manera que en estos casos podríamos sacar factor común y expresar   como producto de dos números. Sin embargo, si  , no podremos sacar factor común, de modo que en estos casos r podrá tomar el valor 2,3, que son los primos que no vienen dados por el reloj.

Por tanto, todo número primo en este Reloj, puede expresarse de la forma aquí establecida.

2

d) y a las mismas 6 horas aparecen todos.e) los intervalos del reloj (en el primer giro de su aguja) son los siguientes:

4, 2, 4, 2, 4, 2.

3

f) la suma de los dígitos de cada uno de los números primos superpuestos, se transforma en el número inicial de cada superposición.

4

g) análisis de los primeros 10000 números primos.

19, 37, 73, 109, 127, 163, 181, 199, 271, 307, 379, 397, 433, 487, 523, 541, 577, 613, 631, 739, 757, 811, 829, 883, 919, 937, 991, 1009 = 1

(2)

(3)

23, 41, 59, 113, 131, 149, 167, 239, 257, 293, 311, 347, 383, 401, 419, 491, 509, 563, 599, 617, 653, 743, 761, 797, 887, 941, 977 = 5

43, 61, 79, 97, 151, 223, 241, 277, 313, 331, 349, 367, 421, 439, 457, 547, 601, 619, 709, 727, 853, 907, 997 = 7

29, 47, 83, 101, 137, 173, 191, 227, 263, 281, 317, 353, 389, 443, 461, 479, 569, 587, 641, 659, 677, 821, 839, 857, 911, 929, 947, 983 = 11

31, 67, 103, 139, 157, 193, 211, 229, 283, 337, 373, 409, 463, 499, 571, 607, 643, 661, 733, 751, 769, 787,823, 859, 877, 967 = 13

53, 71, 89, 107, 179, 197, 233, 251, 269, 359, 431, 449, 467, 503, 521, 557, 593, 647, 683, 701, 719, 773, 809, 827, 863, 881, 953, 971 = 17

1009 = 10 = 11013 = 51019 = 111021 = 4 = 131031 = 51033 = 71039 = 131049 = 14 = 51051 = 71061 = 8 = 171063 = 10 = 11069 = 16 = 71087 = 16 = 71091 = 111093 = 131097 = 171103 = 51109 = 111117 = 10 = 11123 = 71129 = 131151 = 8 = 171153 = 10 = 11163 = 111171 = 10 = 11181 = 111187 = 171193 = 14 = 51201 = 4 = 131213 = 7

5

1217 = 111223 = 8 = 171229 = 14 = 51231 = 71237 = 131249 = 16 = 71259 = 171277 = 171279 = 19 = 11283 = 14 = 51289 = 20 = 2 = 111291 = 131297 = 19 = 11301 = 51303 = 71307 = 111319 = 14 = 51321 = 71327 = 131361 = 111367 = 171373 = 14 = 51381 = 131399 = 22 = 4 = 131409 = 131423 = 10 = 11427 = 14 = 51429 = 16 = 71433 = 111439 = 171447 = 16 = 71451 = 111453 = 14 = 51459 = 19 = 11471 = 131481 = 14 = 51483 = 16 = 71487 = 20 = 2 = 111489 = 22 = 4 = 131493 = 171499 = 23 = 51511 = 8 = 171523 = 111531 = 10 = 11543 = 131549 = 19 = 11553 = 14 = 51559 = 20 = 2 = 111567 = 19 = 11571 = 14 = 51579 = 22 = 4 = 131583 = 171597 = 22 = 4 = 131601 = 8 = 171607 = 14 = 5

6

1609 = 16 = 71613 = 111619 = 171621 = 10 = 11627 = 16 = 71637 = 171657 = 19 = 11663 = 16 = 71667 = 20 = 2 = 111669 = 22 = 4 = 131693 = 19 = 11697 = 23 = 51699 = 25 = 71709 = 171721 = 111723 = 131733 = 14 = 51741 = 131747 = 19 = 11753 = 16 = 71759 = 22 = 4 = 131777 = 22 = 4 = 131783 = 19 = 11787 = 23 = 51789 = 25 = 71801 = 10 = 11811 = 111823 = 14 = 51831 = 131847 = 20 = 4 = 131861 = 16 = 71867 = 22 = 4 = 131871 = 171873 = 19 = 11877 = 23 = 51879 = 25 = 71889 = 26 = 8 = 171901 = 111907 = 171913 = 14 = 51931 = 14 = 51933 = 16 = 71949 = 23 = 51951 = 16 = 71973 = 20 = 2 = 111979 = 26 = 8 = 171987 = 25 = 71993 = 22 = 4 = 131997 = 26 = 8 = 171999 = 28 = 10 = 12003 = 52011 = 4 = 132017 = 10 = 12027 = 112029 = 13

7

2039 = 14 = 52053 = 10 = 12063 = 112069 = 172081 = 112083 = 132087 = 172089 = 19 = 12099 = 20 = 2 = 112111 = 52113 = 72129 = 14 = 52131 = 72137 = 132141 = 8 = 172143 = 10 = 12153 = 112161 = 10 = 12179 = 19 = 12203 = 72207 = 112213 = 8 = 172221 = 72237 = 14 = 52239 = 16 = 72243 = 112251 = 102267 = 172269 = 19 = 12273 = 14 = 52281 = 132287 = 19 = 12293 = 16 = 72297 = 20 = 2 = 112309 = 14 = 52311 = 72333 = 112339 = 172341 = 10 = 12347 = 16 = 72351 = 112357 = 172371 = 132377 = 19 = 12381 = 14 = 52383 = 16 = 72389 = 22 = 4 = 132393 = 172399 = 23 = 52411 = 8 = 172417 = 14 = 52423 = 112437 = 16 = 72441 = 112447 = 17

8

2459 = 20 = 2 = 112467 = 19 = 12473 = 16 = 72477 = 20 = 2 = 112503 = 10 = 12521 = 10 = 12531 = 112539 = 19 = 12543 = 14 = 52549 = 20 = 2 = 112551 = 132557 = 19 = 12579 = 23 = 52591 = 172593 = 19 = 12609 = 172617 = 16 = 72621 = 112633 = 14 = 52647 = 19 = 12657 = 20 = 2 = 112659 = 22 = 4 = 132663 = 172671 = 16 = 72677 = 22 = 4 = 132683 = 19 = 12687 = 23 = 52689 = 25 = 72693 = 20 = 2 = 112699 = 26 = 8 = 172707 = 16 = 72711 = 112713 = 132719 = 19 = 12729 = 20 = 2 = 112731 = 132741 = 14 = 52749 = 22 = 4 = 132753 = 172767 = 22 = 4 = 132777 = 23 = 52789 = 26 = 8 = 172791 = 19 = 12797 = 25 = 72801 = 112803 = 132819 = 20 = 2 = 112833 = 16 = 72837 = 20 = 2 = 112843 = 172851 = 16 = 72857 = 22 = 4 = 132861 = 172879 = 26 = 8 = 172887 = 25 = 7

9

2897 = 26 = 8 = 172903 = 14 = 52909 = 20 = 2 = 112917 = 19 = 12927 = 20 = 2 = 112939 = 23 = 52953 = 19 = 12957 = 23 = 52963 = 20 = 2 = 112969 = 26 = 8 = 172971 = 19 = 12999 = 29 = 113001 = 4 = 133011 = 53019 = 133023 = 6 = 173037 = 133041 = 8 = 173049 = 16 = 73061 = 10 = 13067 = 16 = 73079 = 19 = 13083 = 14 = 53089 = 20 = 2 = 113109 = 133119 = 14 = 53121 = 73137 = 14 = 53163 = 133167 = 173169 = 19 = 13181 = 133187 = 19 = 13191 = 14 = 53203 = 8 = 173209 = 14 = 53217 = 133221 = 8 = 173229 = 16 = 73251 = 113253 = 133257 = 173259 = 20 = 2 = 113271 = 133299 = 23 = 53301 = 73307 = 133313 = 10 = 13319 = 16 = 73323 = 113329 = 173331 = 10 = 13343 = 133347 = 173359 = 20 = 2 = 11

10

3361 = 133371 = 14 = 53373 = 16 = 73389 = 23 = 53391 = 16 = 73407 = 14 = 53413 = 113433 = 133449 = 20 = 2 = 113457 = 19 = 13461 = 14 = 53463 = 16 = 73467 = 20 = 2 = 113469 = 22 = 4 = 133491 = 173499 = 25 = 73511 = 10 = 13517 = 16 = 73527 = 173529 = 19 = 13533 = 14 = 53539 = 20 = 2 = 113541 = 133547 = 19 = 13557 = 20 = 2 = 113559 = 22 = 4 = 133571 = 16 = 73581 = 173583 = 19 = 13593 = 20 = 2 = 113607 = 16 = 73613 = 133617 = 173623 = 14 = 53631 = 133637 = 19 = 13643 = 16 = 73659 = 23 = 53671 = 173673 = 19 = 13677 = 23 = 53691 = 19 = 13697 = 25 = 73701 = 113709 = 19 = 13719 = 20 = 2 = 113727 = 19 = 13733 = 16 = 73739 = 22 = 4 = 133761 = 173767 = 23 = 53769 = 25 = 73779 = 26 = 8 = 173793 = 22 = 4 = 133797 = 26 = 8 = 17

11

3803 = 14 = 53821 = 14 = 53823 = 16 = 73833 = 173847 = 22 = 4 = 133851 = 173853 = 19 = 13863 = 20 = 2 = 113877 = 25 = 73881 = 20 = 113889 = 28 = 10 = 13907 = 19 = 13911 = 14 = 53917 = 20 = 2 = 113919 = 22 = 4 = 133923 = 173929 = 23 = 53931 = 16 = 73943 = 19 = 13947 = 23 = 53967 = 25 = 73989 = 29 = 114001 = 54003 = 74007 = 114013 = 8 = 174019 = 14 = 54021 = 74027 = 134049 = 174051 = 10 = 14057 = 16 = 74073 = 14 = 54079 = 20 = 2 = 114091 = 14 = 54093 = 16 = 74099 = 22 = 4 = 134111 = 74127 = 14 = 54129 = 16 = 74133 = 114139 = 174153 = 134157 = 174159 = 19 = 14177 = 19 = 14201 = 74211 = 8 = 174217 = 14 = 54219 = 16 = 74229 = 174231 = 10 = 14241 = 114243 = 134253 = 14 = 5

12

4259 = 20 = 2 = 114261 = 134271 = 14 = 54273 = 16 = 74283 = 174289 = 23 = 54297 = 22 = 4 = 134327 = 16 = 74337 = 174339 = 19 = 14349 = 20 = 2 = 114357 = 19 = 14363 = 16 = 74373 = 174391 = 174397 = 22 = 4 = 134409 = 174421 = 114423 = 134441 = 134447 = 19 = 14451 = 14 = 54457 = 20 = 2 = 114463 = 174481 = 174483 = 19 = 14493 = 20 = 2 = 114507 = 16 = 74513 = 134517 = 174519 = 19 = 14523 = 14 = 54547 = 20 = 2 = 114549 = 22 = 4 = 134561 = 16 = 74567 = 22 = 4 = 134583 = 20 = 2 = 114591 = 19 = 14597 = 25 = 74603 = 134621 = 134637 = 20 = 2 = 114639 = 22 = 4 = 134643 = 174649 = 23 = 54651 = 16 = 74657 = 22 = 4 = 134663 = 19 = 14673 = 20 = 2 = 114679 = 26 = 8 = 174691 = 20 = 2 = 114703 = 14 = 54721 = 14 = 54723 = 16 = 74729 = 22 = 4 = 13

13

4733 = 174751 = 174759 = 25 = 74783 = 22 = 4 = 134787 = 26 = 8 = 174789 = 28 = 10 = 14793 = 23 = 54799 = 29 = 114801 = 134813 = 16 = 74817 = 20 = 2 = 114831 = 16 = 74861 = 19 = 14871 = 20 = 2 = 114877 = 26 = 8 = 174889 = 29 = 114903 = 16 = 74909 = 22 = 4 = 134919 = 23 = 54931 = 174933 = 19 = 14937 = 23 = 54943 = 20 = 2 = 114951 = 19 = 14957 = 25 = 74967 = 26 = 8 = 174969 = 28 = 10 = 14973 = 23 = 54987 = 28 = 10 = 14993 = 25 = 74999 = 31 = 4 = 135003 = 8 = 175009 = 14 = 55011 = 75021 = 8 = 175023 = 10 = 15039 = 175051 = 115059 = 19 = 15077 = 19 = 15081 = 14 = 55087 = 20 = 2 = 115099 = 23 = 55101 = 75107 = 135113 = 10 = 15119 = 16 = 75147 = 175153 = 14 = 55167 = 19 = 15171 = 14 = 55179 = 22 = 4 = 135189 = 23 = 55197 = 22 = 4 = 135209 = 16 = 7

14

5227 = 16 = 75231 = 115233 = 135237 = 175261 = 14 = 55273 = 175279 = 23 = 55281 = 16 = 75297 = 23 = 55303 = 115309 = 175323 = 135333 = 14 = 55347 = 19 = 15351 = 14 = 55381 = 175387 = 23 = 55393 = 20 = 2 = 115399 = 26 = 8 = 175407 = 16 = 75413 = 135417 = 175419 = 19 = 15431 = 135437 = 19 = 15441 = 14 = 55443 = 16 = 75449 = 22 = 4 = 135471 = 175477 = 23 = 55479 = 25 = 75483 = 20 = 2 = 115501 = 115503 = 135507 = 175519 = 20 = 2 = 115521 = 135527 = 19 = 15531 = 14 = 55557 = 22 = 4 = 135563 = 19 = 15569 = 25 = 75573 = 20 = 2 = 115581 = 19 = 15591 = 20 = 2 = 115623 = 16 = 75639 = 23 = 55641 = 16 = 75647 = 22 = 4 = 135651 = 175653 = 19 = 15657 = 23 = 55659 = 25 = 75669 = 26 = 8 = 175683 = 22 = 4 = 13

15

5689 = 28 = 10 = 15693 = 23 = 55701 = 135711 = 14 = 55717 = 20 = 2 = 115737 = 22 = 4 = 135741 = 175743 = 19 = 15749 = 25 = 75779 = 28 = 10 = 15783 = 23 = 55791 = 22 = 4 = 135801 = 14 = 55807 = 20 = 2 = 115813 = 175821 = 16 = 75827 = 22 = 4 = 135839 = 25 = 75843 = 20 = 2 = 115849 = 26 = 8 = 175851 = 19 = 15857 = 25 = 75861 = 20 = 2 = 115867 = 26 = 8 = 175869 = 28 = 10 = 15879 = 29 = 115881 = 22 = 4 = 135897 = 29 = 115903 = 175923 = 19 = 15927 = 23 = 55939 = 26 = 8 = 175953 = 22 = 4 = 135981 = 23 = 55987 = 29 = 116007 = 136011 = 8 = 176029 = 176037 = 16 = 76043 = 13 6047 = 176053 = 14 = 56067 = 19 = 16073 = 16 = 76079 = 22 = 4 = 136089 = 23 = 56091 = 16 = 76101 = 8 = 176113 = 116121 = 10 = 16131 = 116133 = 136143 = 14 = 56151 = 136163 = 16 = 7

16

6173 = 176197 = 23 = 56199 = 25 = 76203 = 116211 = 10 = 16217 = 16 = 76221 = 116229 = 19 = 16247 = 19 = 16257 = 20 = 2 = 116263 = 176269 = 23 = 56271 = 16 = 76277 = 22 = 4 = 136287 = 23 = 56299 = 26 = 8 = 176301 = 10 = 16311 = 116317 = 176323 = 14 = 56329 = 20 = 2 = 116337 = 19 = 16343 = 16 = 76353 = 176359 = 23 = 56361 = 16 = 76367 = 22 = 4 = 136373 = 19 = 16379 = 25 = 76389 = 26 = 8 = 176397 = 25 = 76421 = 136427 = 19 = 16449 = 23 = 56451 = 16 = 76469 = 25 = 76473 = 20 = 2 = 116481 = 19 = 16491 = 20 = 2 = 116521 = 14 = 56529 = 22 = 4 = 136547 = 22 = 4 = 136551 = 176553 = 19 = 16563 = 20 = 2 = 116569 = 26 = 8 = 176571 = 19 = 16577 = 25 = 76581 = 20 = 2 = 116599 = 29 = 116607 = 19 = 16619 = 22 = 4 = 136637 = 22 = 4 = 136653 = 20 = 2 = 116659 = 26 = 8 = 17

17

6661 = 19 = 16673 = 22 = 4 = 136679 = 28 = 10 = 16689 = 29 = 116691 = 22 = 4 = 136701 = 14 = 56703 = 16 = 76709 = 22 = 4 = 136719 = 23 = 56733 = 19 = 16737 = 23 = 56761 = 20 = 2 = 116763 = 22 = 4 = 136779 = 29 = 116781 = 22 = 136791 = 23 = 56793 = 25 = 76803 = 176823 = 19 = 16827 = 23 = 56829 = 25 = 76833 = 20 = 2 = 116841 = 19 = 16857 = 26 = 8 = 176863 = 23 = 56869 = 29 = 116871 = 22 = 4 = 136883 = 25 = 76899 = 32 = 56907 = 22 = 4 = 136911 = 176917 = 23 = 56947 = 26 = 8 = 176949 = 28 = 10 = 16959 = 29 = 116961 = 22 = 4 = 136967 = 28 = 10 = 16971 = 23 = 56977 = 29 = 116983 = 26 = 8 = 176991 = 25 = 76997 = 31 = 4 = 137001 = 8 = 177013 = 117019 = 177027 = 16 = 77039 = 19 = 17043 = 14 = 57057 = 19 = 17069 = 22 = 4 = 137079 = 23 = 57103 = 117109 = 177121 = 117127 = 17

18

7129 = 19 = 17151 = 14 = 57159 = 22 = 4 = 137177 = 22 = 4 = 137187 = 23 = 57193 = 20 = 2 = 117207 = 16 = 77211 = 117213 = 137219 = 19 = 17229 = 20 = 2 = 117237 = 19 = 17243 = 16 = 77247 = 20 = 2 = 117253 = 177283 = 20 = 2 = 117297 = 25 = 77307 = 177309 = 19 = 17321 = 137331 = 14 = 57333 = 16 = 77349 = 23 = 57351 = 16 = 77369 = 25 = 77393 = 22 = 4 = 137411 = 137417 = 19 = 17433 = 177451 = 177457 = 23 = 57459 = 25 = 77477 = 25 = 77481 = 20 = 2 = 117487 = 26 = 8 = 177489 = 28 = 19 = 17499 = 29 = 117507 = 19 = 17517 = 20 = 2 = 117523 = 177529 = 23 = 57537 = 22 = 4 = 137541 = 177547 = 23 = 57549 = 25 = 77559 = 26 = 8 = 177561 = 19 = 17573 = 22 = 4 = 137577 = 26 = 8 = 177583 = 23 = 57589 = 29 = 117591 = 22 = 4 = 137603 = 16 = 77607 = 20 = 2 = 117621 = 16 = 7

19

7639 = 25 = 77643 = 20 = 2 = 117649 = 26 = 8 = 177669 = 28 = 10 = 17673 = 23 = 57681 = 22 = 4 = 137687 = 28 = 10 = 17691 = 23 = 57699 = 31 = 4 = 137703 = 177717 = 22 = 4 = 137723 = 19 = 17727 = 23 = 57741 = 19 = 17753 = 22 = 4 = 137757 = 26 = 8 = 177759 = 28 = 10 = 17789 = 31 = 4 = 137793 = 26 = 8 = 177817 = 23 = 57823 = 20 = 2 = 117829 = 26 = 8 = 177841 = 20 = 2 = 117853 = 23 = 57867 = 28 = 10 = 17873 = 25 = 77877 = 29 = 117879 = 31 = 4 = 137883 = 26 = 8 = 177901 = 177907 = 23 = 57919 = 26 = 8 = 177927 = 25 = 77933 = 22 = 4 = 137937 = 26 = 8 = 177949 = 29 = 117951 = 22 = 4 = 137963 = 25 = 77993 = 28 = 10 = 18009 = 178011 = 10 = 18017 = 16 = 78039 = 20 = 2 = 118053 = 16 = 78059 = 22 = 4 = 138069 = 23 = 58081 = 178087 = 23 = 58089 = 25 = 78093 = 20 = 2 = 118101 = 10 = 18111 = 118117 = 178123 = 14 = 58147 = 20 = 2 = 11

20

8161 = 16 = 78167 = 22 = 4 = 138171 = 178179 = 25 = 78191 = 19 = 18209 = 19 = 18219 = 20 = 2 = 118221 = 138231 = 14 = 58233 = 16 = 78237 = 20 = 2 = 118243 = 178263 = 19 = 18269 = 25 = 78273 = 20 = 2 = 118287 = 25 = 78291 = 20 = 2 = 118293 = 22 = 4 = 138297 = 26 = 8 = 178311 = 138317 = 19 = 18329 = 22 = 4 = 138353 = 19 = 18363 = 20 = 2 = 118369 = 26 = 8 = 178377 = 25 = 78387 = 26 = 8 = 178389 = 28 = 10 = 18419 = 22 = 4 = 138423 = 178429 = 23 = 58431 = 16 = 78443 = 19 = 18447 = 23 = 58461 = 19 = 18467 = 25 = 78501 = 14 = 58513 = 178521 = 16 = 78527 = 22 = 4 = 138537 = 23 = 58539 = 25 = 78543 = 20 = 2 = 118563 = 22 = 4 = 138573 = 23 = 58581 = 22 = 4 = 138597 = 29 = 118599 = 31 = 4 = 138609 = 23 = 58623 = 19 = 18627 = 23 = 58629 = 25 = 78641 = 19 = 18647 = 25 = 78663 = 23 = 5

21

8669 = 29 = 118677 = 28 = 10 = 18681 = 23 = 58689 = 31 = 4 = 138693 = 26 = 8 = 178699 = 32 = 58707 = 22 = 4 = 138713 = 19 = 18719 = 25 = 78731 = 19 = 18737 = 25 = 78741 = 20 = 2 = 118747 = 26 = 8 = 178753 = 23 = 58761 = 22 = 4 = 138779 = 31 = 4 = 138783 = 26 = 8 = 178803 = 19 = 18807 = 23 = 58819 = 26 = 8 = 178821 = 19 = 18831 = 20 = 2 = 118837 = 26 = 8 = 178839 = 28 = 10 = 18849 = 29 = 118861 = 23 = 58863 = 25 = 78867 = 29 = 118887 = 31 = 4 = 138893 = 28 = 10 = 18923 = 22 = 4 = 138929 = 28 = 10 = 18933 = 23 = 58941 = 22 = 4 = 138951 = 23 = 58963 = 26 = 8 = 178969 = 32 = 58971 = 25 = 78999 = 35 = 8 = 179001 = 10 = 19007 = 16 = 79011 = 119013 = 139029 = 20 = 2 = 119041 = 14 = 59043 = 16 = 79049 = 22 = 4 = 139059 = 23 = 59067 = 22 = 4 = 139091 = 19 = 19103 = 139109 = 19 = 19127 = 19 = 19133 = 16 = 79137 = 20 = 2 = 11

22

9151 = 16 = 79157 = 22 = 4 = 139161 = 179173 = 20 = 2 = 119181 = 19 = 19187 = 25 = 79199 = 28 = 10 = 19203 = 14 = 59209 = 20 = 2 = 119221 = 14 = 59227 = 20 = 2 = 119239 = 23 = 59241 = 16 = 79257 = 23 = 59277 = 25 = 79281 = 20 = 2 = 119283 = 22 = 4 = 139293 = 23 = 59311 = 14 = 59319 = 22 = 4 = 139323 = 179337 = 22 = 4 = 139341 = 179343 = 19 = 19349 = 25 = 79371 = 20 = 2 = 119377 = 26 = 8 = 179391 = 22 = 4 = 139397 = 28 = 10 = 19403 = 16 = 79413 = 179419 = 23 = 59421 = 16 = 79431 = 179433 = 19 = 19437 = 23 = 59439 = 25 = 79461 = 20 = 2 = 119463 = 22 = 4 = 139467 = 26 = 8 = 179473 = 23 = 59479 = 29 = 119491 = 23 = 59497 = 29 = 119511 = 16 = 79521 = 179533 = 20 = 2 = 119539 = 26 = 8 = 179547 = 25 = 79551 = 20 = 2 = 119587 = 29 = 119601 = 16 = 79613 = 19 = 19623 = 20 = 2 = 119629 = 26 = 8 = 17

23

9631 = 19 = 19643 = 22 = 4 = 139649 = 28 = 10 = 19661 = 22 = 4 = 139677 = 29 = 119679 = 31 = 4 = 139689 = 32 = 59697 = 31 = 4 = 139719 = 26 = 8 = 179721 = 19 = 19733 = 22 = 4 = 139739 = 28 = 10 = 19743 = 23 = 59749 = 29 = 119767 = 29 = 119769 = 31 = 4 = 139781 = 25 = 79787 = 31 = 4 = 139791 = 26 = 8 = 179803 = 20 = 2 = 119811 = 19 = 19817 = 25 = 79829 = 28 = 10 = 19833 = 23 = 59839 = 29 = 119851 = 23 = 59857 = 29 = 119859 = 31 = 4 = 139871 = 25 = 79883 = 28 = 10 = 19887 = 32 = 59901 = 19 = 19907 = 25 = 79923 = 23 = 59929 = 29 = 119931 = 22 = 4 = 139941 = 23 = 59949 = 31 = 4 = 139967 = 31 = 4 = 139973 = 28 = 10 = 110007 = 8 = 17

h) análisis de números primos más grandes.

1299709, 37124508045065437, 44211790234832169331,

170141183460469231731687303715884105727,

20988936657440586486151264256610222593863921… = 1 104729, 22801763489… = 5

179424673… = 7

2038074743… = 11

24

15485863, 2760727302517, 394906913903735329,

357686312646216567629137… = 13

7919, 252097800623, 2999224275833, 323780508946331,

3475385758524527, 4185296581467695669,

13169525310647365859… = 17

1.475.979.915.214.180.235.084.898.622.737.381.736.312.066.145.333.169.775.147.771.216.478.570.297.878.078.949.377.407.337.049.312.066.145.333.169.775.147.771.216.478.570.297.878.078.949.377.407.337.049.389.289.382.748.507.531.496.480.477.281.264.838.760.259.191.814.463.365.330.389.289.382.748.507.531.496.480.477.281.264.838.760.259.191.814.463.365.330.269.540.496.961.201.113.430.156.902.396.093.989.090.226.259.326.935.025.281.269.540.496.961.201.113.430.156.902.396.093.989.090.226.259.326.935.025.281.409.614.983.499.388.222.831.448.598.601.834.318.536.230.923.772.641.390.209.409.614.983.499.388.222.831.448.598.601.834.318.536.230.923.772.641.390.209.490.231.836.446.899.608.210.795.482.963.763.094.236.630.945.410.832.793.769.490.231.836.446.899.608.210.795.482.963.763.094.236.630.945.410.832.793.769.905.399.982.457.186.322.944.729.636.418.890.623.372.171.723.742.105.636.440.905.399.982.457.186.322.944.729.636.418.890.623.372.171.723.742.105.636.440.368.218.459.649.632.948.538.696.905.872.650.486.914.434.637.457.507.280.441.368.218.459.649.632.948.538.696.905.872.650.486.914.434.637.457.507.280.441.823.676.813.517.852.099.348.660.847.172.579.408.422.316.678.097.670.224.011.823.676.813.517.852.099.348.660.847.172.579.408.422.316.678.097.670.224.011.990.280.170.474.894.487.426.924.742.108.823.536.808.485.072.502.240.519.452.990.280.170.474.894.487.426.924.742.108.823.536.808.485.072.502.240.519.452.587.542.875.349.976.558.572.670.229.633.962.575.212.637.477.897.785.501.552.587.542.875.349.976.558.572.670.229.633.962.575.212.637.477.897.785.501.552.646.522.609.988.869.914.013.540.483.809.865.681.250.419.497.686.697.771.007646.522.609.988.869.914.013.540.483.809.865.681.250.419.497.686.697.771.007 = 6037 = 16 = 7

5.219.283.133.417.550.597.608.921.138.394.131. 714.748.003.987.111.696.388.844.721.857.021.695.621.345.566.328.693.730.284.546.120.701.185.550.350.229.748.838.662.252.951.341.253.421.746.795.073.087.191.036.889.245.882.078.365.188.808.677.470.720.357.783.212.747.307.528.673.196.680.088.458.323.693.474.572.645.646.430.349.088.271.293.578.783.855.910.006.177.172.218.945.638.975.265.804.307.407.923.864.296.665.887.203.966.818.881.644.127.188.326.278.536.823.185.302.433.682.452.220.866.505.473.927.329.966.261.635.224.705.976.307.336.848.031.449.051.946.470.250.326.355.850.564.065.070.219.415.063.390.427.891.641.945.141.848.406.032.059.956.642.759.451.697.871.131.937.435.974.734.468.812.612.404.366.795.155.477.504.403.442.852.091.868.360.651.937.341.101.803.746.125.038.414.968.100.853.875.769.649.772.568.976.533.115.626.733.686.633.581.147.178.167.854.537.374.386.645.054.536.421.581.487.787.680.673.873.260.827.329.457.946.477.205.293.076.461.503.204.195.229.898.682.736.527.831.015.184.366.661.152.864.587.615.185.846.767.471.910.458.921.833.277.044.360.716.991.019.765.678.686.642.856.418.011.096.226.793.070.226.165.298.245.200.286.805.907.696.043.158.401.359.233.205.933.894.872.270.757.840.780.311.842.993.502.030.651.236.226.420.390.626.397.948.002.732.496.153.425.437.136.287.482.563.891.064.276.318.845.679.420.730.548.518.354.628.798.521.087.698.057.831.570.193.388.325.181.647.160.191.379.350.631.674.091.897.212.262.818.278.445.060.746.814.976.957.862.342.535.407.548.922.

25

347.134.501.697.060.190.457.417.366.250.455.037.928.704.152.728.620.313.194.049.459.090.938.561.074.166.466.321.977.154.209.854.449.604.435.530.303.555.162.486.890.044.423.150.383.891.465.784.220.859.099.339.814.855.122.982.824.292.907.630.042.351.718.406.860.216.298.441.018.976.432.899.012.907.690.738.344.741.325.809.832.514.370.637.919.675.751.564.228.983.581.067.713.001.959.082.281.741.038.000.510.382.863.445.395.122.241.581.297.039.798.297.231.129.157.645.252.235.222.643.331.292.666.424.360.475.348.246.102.972.693.436.258.291.783.024.959.062.081.981.838.812.572.142.511.803.956.525.449.888.847.112.458.483.851.365.748.838.790.507.346.393.979.150.203.170.197.392.307.844.865.039.294.565.185.258.407.625.913.055.060.713.900.922.073.408.886.771.350.586.379.576.558.646.737.634.415.678.011.238.370.510.372.009.244.595.665.266.604.273.647.744.438.936.460.505.842.615.012.921.299.723.039.609.880.229.752.363.633.215.766.432.725.435.987.253.902.189.301.146.196.983.131.735.602.940.755.230.848.663.445.117.927.665.658.368.356.420.346.865.950.111.317.140.432.850.742.151.588.086.112.978.702.994.240.762.421.583.385.188.191.350.001.231.200.565.700.667.791.389.021.354.205.858.177.467.953.946.405.058.508.219.437.914.852.128.789.874.720.946.029.689.233.060.816.869.125.386.142.828.340.576.189.232.460.988.484.979.947.357.301.166.335.302.699.325.444.851.058.733.209.447.959.819.326.191.635.260.772.935.604.990.851.134.296.788.845.298.869.594.134.244.897.534.525.730.181.304.428.393.857.426.219.940.590.672.281.276.723.518.776.582.169.544.067.334.705.480.920.742.604.001.168.622.762.198.273.693.428.380.553.897.726.338.361.872.954.339.053.214.349.828.778.842.361.781.853.106.705.246.246.112.826.667.187.778.644.452.294.726.844.594.574.918.377.285.572.198.422.309.281.050.239.527.692.661.557.089.591.842.761.278.181.057.933.278.000.401.648.351.356.081.767.219.918.593.264.919.576.995.069.142.249.170.223.836.224.702.255.068.465.253.482.078.539.098.187.443.071.918.215.944.764.927.958.379.845.359.365.240.314.888.899.686.669.959.635.761.817.082.650.578.399.550.074.453.202.478.150.863.419.553.174.171.457.756.089.567.615.842.152.519.360.157.327.308.360.698.725.072.256.365.040.969.792.621.539.584.036.465.460.483.849.963.364.877.060.077.099.462.804.508.783.747.443.748.521.994.786.797.187.208.616.764.402.042.119.167.359.653.346.884.051.067.448.362.794.008.940.082.341.100.994.444.181.614.418.213.550.484.684.610.370.257.475.208.798.374.548.203.571.375.481.853.250.719.794.707.737.297.289.106.817.344.273.543.178.234.366.780.318.872.649.407.377.187.072.730.875.789.746.785.084.870.361.011.603.837.828.596.063.380.436.934.427.559.741.399.079.630.414.229.895.319.266.149.914.612.519.280.272.823.995.869.813.896.972.841.782.561.116.929.003.166.739.744.612.062.920.048.993.540.969.166.539.567.069.660.783.098.613.110.181.470.263.393.931.915.610.974.133.757.611.745.434.282.513.504.673.223.466.819.904.565.802.892.335.444.902.334.736.162.918.420.861.856.512.137.424.964.551.732.430.578.193.645.849.344.558.708.572.144.258.113.789.862.433.670.385.490.704.828.913.699.643.122.856.631.354.937.003.427.216.841.447.763.823.770.059.461.987.956.351.233.769.350.148.968.100.975.738.387.546.952.316.818.146.789.228.505.502.633.995.283.191.008.860.757.224.878.910.306.387.822.619.841.228.487.510.959.391.915.230.081.278.738.478.964.587.060.771.579.566.927.543.809.518.001.023.256.799.064.843.500.598.047.448.092.769.672.112.496.204.842.133.809.466.280.596.134.763.991.644.871.105.523.092.033.002.144.386.358.014.252.471.014.304.933.512.315.525.855.464.929.326.002.127.213.531.185.766.066.959.495.012.425.333.202.747.713.991.260.204.270.554.711.267.603.519.448.421.573.364.790.583.464.599.414.079.092.630.562.559.049.529.970.854.068.245.838.058.972.902.782.716.657.303.398.011.079.570.340.993.681.978.602.146.016.260.731.415.392.796.070.099.302.915.857.446.184.284.492.338.184.900.216.275.081.239.194.286.014.920.899.856.256.657.301.752.101.626.142.771.298.248.388.740.363.577.822.427.523.530.496.698.324.976.089.830.267.271.770.433.302.612.928.863.492.471.906.219.529.755.857.499.316.371.756.474.544.420.059.869.354.061.575.644.223.098.201.545.184.395.439.200.495.673.217.450.313.417.117.617.847.652.

26

221.602.698.562.899.339.244.049.925.561.342.019.071.478.094.656.156.692.298.970.839.460.894.861.033.151.782.740.390.479.827.962.701.287.562.750.064.371.964.027.625.677.232.269.761.059.437.891.312.572.243.255.115.277.459.236.700.764.735.484.999.611.933.639.503.357.777.150.911.412.524.162.248.723.431.001.175.341.571.889.620.281.950.611.889.360.490.007.342.476.142.047.713.725.454.664.181.989.231.971.490.749.598.648.207.558.524.714.435.823.528.801.963.270.106.690.871.228.191.029.010.653.103.641.813.849.875.532.556.927.413.266.316.430.858.480.756.663.121.051.354.045.742.088.249.205.913.851.178.254.507.537.826.478.453.318.305.024.009.219.910.295.626.907.315.048.440.247.469.362.688.952.564.187.604.507.596.955.998.539.879.540.191.650.667.035.389.903.303.539.491.945.912.991.638.480.371.092.660.980.900.779.331.006.985.407.097.736.196.322.796.802.626.314.674.103.633.849.922.510.328.783.478.267.810.901.914.263.565.926.331.779.841.234.341.557.171.622.225.997.181.620.978.940.634.079.517.600.724.299.703.807.480.826.218.272.492.787.987.996.449.775.919.730.062.045.133.756.639.675.098.203.801.102.844.045.509.184.301.293.110.435.898.724.583.832.877.280.053.291.902.793.566.611.159.694.010.696.228.754.063.771.664.996.095.787.812.423.235.160.254.094.397.254.125.391.108.561.499.107.218.473.169.425.539.288.882.278.513.046.714.043.407.319.219.068.949.468.172.288.158.436.366.324.718.101.955.588.003.305.237.184.455.698.919.819.573.905.719.538.347.163.800.330.464.965.350.384.709.792.997.192.244.756.941.845.904.840.558.998.378.244.156.887.880.300.448.205.407.800.127.556.895.875.369.630.084.390.234.520.371.765.003.246.158.785.015.315.017.571.785.917.762.532.744.084.067.842.701.965.827.402.808.244.353.884.339.008.814.967.552.009.585.707.585.930.199.840.219.015.697.516.779.314.043.811.604.512.416.068.649.879.615.394.713.114.751.547.609.261.771.572.859.645.987.657.096.712.479.535.823.843.294.183.064.535.902.637.275.082.610.415.092.299.017.574.941.271.253.130.505.203.786.422.599.954.246.588.012.241.306.721.923.065.806.890.956.746.479.029.456.426.379.831.947.739.549.627.656.030.524.765.928.950.331.915.786.548.762.562.328.083.171.649.815.759.067.667.516.704.588.106.333.557.929.886.109.775.405.451.970.851.524.312.228.546.878.654.197.960.754.013.026.372.859.274.195.306.213.445.266.974.071.935.007.298.641.663.616.314.033.269.884.667.955.620.111.309.575.343.217.861.185.691.394.535.706.369.722.324.626.811.632.890.909.545.866.042.361.012.706.332.668.814.574.002.886.089.482.457.047.069.792.424.592.306.019.142.598.077.930.541.370.804.870.536.746.764.718.703.756.101.315.103.402.498.751.465.888.822.984.172.394.408.820.511.781.517.872.764.205.955.480.840.803.205.483.107.196.994.531.153.451.950.583.461.658.019.927.392.514.304.980.398.554.764.126.899.265.779.300.584.663.624.849.476.060.120.748.057.077.601.917.768.935.135.399.413.692.676.935.044.564.082.114.422.782.031.238.459.636.513.796.777.783.240.689.793.652.195.084.844.112.602.537.745.301.670.926.262.492.546.259.891.100.605.861.657.849.178.159.354.817.407.487.478.931.490.801.670.084.160.896.230.614.396.947.358.808.585.239.738.180.232.916.108.965.188.695.590.452.014.440.883.246.282.331.273.025.204.280.241.388.275.626.868.589.146.434.017.242.042.021.840.309.157.931.211.813.867.467.410.616.665.884.021.235.841.658.135.804.262.812.971.569.692.978.840.911.041.081.726.688.997.548.959.744.100.838.476.816.595.900.886.448.055.145.128.729.248.758.017.622.720.446.939.441.868.806.149.463.547.490.282.358.163.060.869.321.029.448.427.336.843.444.204.243.515.131.163.997.504.195.558.101.171.711.507.769.963.992.597.654.450.780.344.736.522.510.720.458.589.263.076.972.478.692.329.843.045.248.108.600.587.281.975.177.627.285.940.306.164.121.502.635.012.077.569.268.382.618.992.424.728.960.577.293.015.052.952.673.828.205.703.560.705.531.570.968.207.693.918.162.579.001.277.852.045.222.671.248.467.264.497.211.694.624.169.858.653.954.833.760.474.356.592.813.229.626.505.890.505.354.134.635.151.787.948.216.365.327.901.508.819.411.785.142.152.833.432.627.728.196.612.826.280.426.493.418.494.354.332.715.759.068.513.275.267.431.843.575.112.419.627.392.320.363.061.901.254.840.884.481.021.434.867.813.860.576.688.519.853.931.319.751.

27

135.758.131.820.799.087.677.142.019.115.811.990.730.366.088.825.026.705.967.895.514.998.294.919.077.502.618.682.136.729.305.438.490.581.592.513.614.486.145.775.140.327.508.036.090.571.478.743.601.469.925.100.309.139.079.297.515.952.651.542.458.232.938.433.515.373.375.950.850.550.901.991.034.330.759.166.084.076.529.135.116.217.558.488.006.169.081.262.614.724.001.416.460.184.550.734.010.510.439.401.941.574.568.085.146.192.143.492.506.415.272.776.443.357.698.527.499.205.920.188.662.762.249.868.506.706.790.255.417.996.527.667.702.229.820.002.517.515.132.500.604.313.453.961.565.148.329.696.168.408.722.070.991.415.891.197.822.983.055.527.178.331.435.752.092.793.749.578.216.888.157.239.353.475.522.775.912.927.356.042.862.965.707.543.262.449.655.486.923.996.924.705.591.337.073.224.376.318.210.126.946.003.029.950.798.609.319.746.198.936.150.871.899.457.221.884.322.874.567.454.461.496.564.928.537.222.435.306.561.085.047.593.752.078.023.600.242.671.756.689.325.995.928.149.581.875.252.520.763.579.817.026.930.285.253.426.477.336.763.345.530.568.689.085.250.180.550.154.101.605.421.383.187.700.964.376.878.757.622.750.533.373.212.825.816.132.162.211.868.283.524.579.681.634.439.593.450.986.867.763.869.643.415.251.323.936.438.759.726.268.396.342.400.872.357.875.606.140.671.113.018.462.893.096.298.958.797.589.424.541.173.503.692.724.334.748.353.008.327.094.914.861.878.511.664.919.938.546.847.178.441.862.054.576.369.254.241.418.859.359.997.673.824.895.122.765.329.123.670.077.022.738.684.049.477.623.013.354.061.028.433.153.254.934.063.535.551.741.212.284.366.260.945.270.521.912.247.754.058.217.884.488.363.961.097.022.478.341.654.879.669.119.039.422.474.711.453.560.713.897.625.651.902.545.874.268.502.591.066.142.164.065.201.145.925.431.736.206.354.259.681.236.123.782.295.953.421.146.036.233.320.706.202.542.802.046.619.108.202.210.601.905.731.146.091.245.445.382.171.357.426.106.435.391.257.241.247.454.462.347.439.843.026.848.957.008.024.971.000.728.739.612.764.701.650.820.745.557.077.521.019.023.663.558.599.300.030.098.767.733.318.296.212.267.791.898.625.397.335.732.188.623.883.018.403.444.800.160.899.513.961.282.647.304.597.132.863.167.285.590.498.796.533.476.782.410.947.610.427.209.426.752.715.165.869.523.201.813.729.432.914.018.824.665.609.813.028.365.558.164.278.232.504.560.433.356.526.239.382.675.492.262.016.251.904.662.875.264.317.012.930.368.279.881.629.876.130.193.702.623.522.145.536.349.509.128.321.152.604.697.440.779.119.762.879.148.757.698.034.629.396.659.742.679.932.515.174.448.457.818.126.556.092.989.541.554.952.290.365.028.495.230.047.528.076.275.510.817.038.546.562.525.047.710.332.555.203.233.199.920.185.022.949.987.410.470.356.466.217.859.084.515.621.715.814.744.078.133.458.401.690.440.973.632.241.126.599.062.439.131.812.445.571.831.195.750.648.033.599.093.956.943.441.851.996.548.154.573.576.590.514.545.685.636.340.512.675.499.356.488.849.184.088.303.003.552.394.888.593.264.605.119.630.946.263.677.978.036.135.351.207.949.364.007.605.467.423.376.526.619.612.619.708.816.827.290.437.234.533.145.243.193.982.617.179.538.944.195.289.996.255.922.150.763.223.450.417.133.426.698.647.254.499.958.391.012.721.079.500.439.885.784.286.798.208.974.008.103.527.525.798.997.123.332.642.177.873.443.192.384.052.822.584.344.480.513.642.088.070.178.308.582.176.337.990.887.892.534.104.789.618.366.717.917.812.438.060.665.661.167.665.404.965.586.160.176.289.888.772.605.859.421.114.049.525.300.201.228.676.436.639.481.076.283.265.730.005.578.214.880.015.011.360.655.245.823.382.454.752.145.003.922.761.176.793.176.860.917.776.210.439.643.878.167.980.131.866.355.801.604.038.300.074.595.892.903.069.711.260.374.423.051.281.344.007.558.766.331.046.287.488.860.890.165.799.881.131.335.225.761.972.320.295.123.203.858.013.644.354.191.760.983.660.411.514.591.556.087.245.015.247.095.059.134.575.699.390.131.848.115.923.393.546.207.714.885.164.019.016.901.853.980.277.478.069.412.283.251.180.591.353.065.417.541.684.186.732.481.744.926.646.830.063.532.770.223.682.901.032.098.939.874.367.730.081.608.222.995.311.565.991.349.593.473.838.913.982.760.300.921.230.171.795.098.158.732.723.221.531.830.432.950.856.290.007.606.569.936.454.419.490.342.239.200.539.

28

463.842.248.283.452.814.424.560.739.976.630.988.284.771.270.307.905.269.366.578.858.096.343.588.951.243.106.238.857.167.947.528.386.703.433.424.496.340.998.917.553.854.024.438.991.607.375.269.389.374.913.330.083.682.670.659.443.407.860.772.695.251.623.499.496.681.365.629.776.020.551.122.942.020.147.967.701.233.170.971.164.565.472.853.581.741.935.610.647.156.261.442.384.055.202.447.461.052.756.324.428.747.528.360.646.913.527.525.663.305.805.317.605.697.845.478.025.727.444.978.208.876.304.016.703.016.714.358.509.239.681.996.268.496.084.710.105.913.501.244.180.545.330.136.150.204.618.138.276.030.314.331.596.601.730.799.483.529.675.130.449.879.705.835.640.253.303.475.624.331.762.638.560.430.115.055.607.878.412.858.576.302.603.986.169.310.524.352.365.499.033.295.536.709.243.931.053.749.941.781.216.666.676.849.722.636.559.670.195.363.085.481.243.708.523.193.553.203.468.045.286.868.687.572.241.116.457.429.815.818.493.904.095.679.641.472.113.827.308.948.644.547.480.831.048.764.793.974.598.219.440.703.597.760.450.979.730.526.784.548.348.528.478.981.126.440.100.335.589.564.511.210.080.718.338.720.668.292.455.434.033.777.745.737.329.876.314.888.817.706.542.072.502.546.494.305.848.810.503.776.519.846.387.216.503.354.243.469.167.270.879.296.292.107.216.458.214.318.938.620.321.916.242.246.419.321.692.446.400.984.188.491.107.920.927.885.540.506.689.869.333.319.550.124.121.531.961.967.637.785.140.171.734.782.180.729.355.318.808.490.350. 500.625.033.283.733.486.196.374.702.998.782.686.801.906.999.697.823.374.757.576.051.529.793.966.335.783.838.480.147.339.455.936.666.307.689.959.754.516.238.886.484.633.922.071.530.920.738.265.836.010.765.063.579.426.399.950.855.264.177.745.045.360.357.798.058.459.894.165.283.550.388.823.443.191.581.207.989.394.041.699.662.793.780.547.315.971.739.049.404.334.692.755.294.151.636.293.340.145.938.553.396.184.505.733.191.406.711.054.032.381.118.628.275.046.047.882.209.953.569.468.206.198.402.245.769.816.997.661.783.644.787.630.668.623.255.824.944.069.180.939.286.643.322.416.802.183.432.233.004.319.271.910.928.887.103.445.885.987.503.538.865.217.204.903.661.862.802.860.195.870.304.417.691.385.203.611.714.502.909.568.732.628.386.772.288.321.101.737.981.952.503.908.657.510.268.626.233.956.746.929.677.891.343.398.693.108.833.172.456.699.933.150.924.423.772.966.255.944.302.389.807.107.679.885.786.647.357.304.745.578.601.085.057.960.865.716.938.291.990.517.461.346.949.345.588.040.786.702.324.417.668.600.590.221.394.530.822.148.873.687.230.167.938.390.325.669.700.283.394.317.526.993.044.736.683.267.934.127.199.505.163.235.323.859.199.411.955.121.008.904.060.213.692.888.039.563.072.698.548.205.248.121.582.170.609.366.676.650.613.275.568.505.188.142.341.951.040.544.825.408.567.652.588.518.558.585.906.517.079.389.070.196.584.208.638.012.431.962.769.864.845.935.249.042.575.015.105.851.544.443.130.170.041.766.567.612.163.646.516.289.495.143.859.625.803.124.769.550.214.130.927.323.677.706.000.126.547.367.711.187.358.932.033.441.900.785.783.804.522.154.484.942.994.157.381.726.864.957.796.374.317.989.124.569.595.601.718.921.105.964.163.751.627.872.221.731.073.259.057.071.671.114.748.468.089.122.507.573.108.262.957.947.467.318.792.776.413.329.324.126.543.145.101.325.421.150.794.742.401.365.275.910.113.372.357.193.932.014.729.097.560.576.009.796.970.589.125.132.018.505.242.435.049.021.999.190.501.788.022.198.184.026.497.465.786.915.101.220.787.507.788.225.473.024. 214.784.688.957.422.903.036.584.053.396.565.930.862.638.025.590.443.803.676.419.182.052.755.763.247.523.593.787.911.111.805.458.919.102.073.103.089.396.698.803.580.832.272.305.421.557.256.437.585.106.474.206.596.838.008.830.418.989.029.710.524.037.994.857.577.266.645.231.673.510.146.636.159.675.360.017.287.419.793.516.512.154.770.352.296.551.724.089.267.870.934.698.157.466.371.337.582.272.478.646.002.708.199.948.393.973.530.096.606.317.370.926.740.805.647.404.584.433.338.728.747.758.787.449.799.483.214.725.657.610.666.333.689.585.296.497.210.167.818.951.260.851.947.688.849.515.831.641.598.466.707.355.270.906.108.846.342.509.978.285.495.939.770.115.709.354.671.057.029.584.411.551.950.522.223.690.988.001.144.258.550.273.342.538.920.445.248.869.080.701.

29

571.113.838.148.084.007.434.013.046.317.661.547.673.470.938.239.992.648.254.419.143.712.021.487.373.234.721.473.431.991.119.240.737.093.415.252.683.167.731.028.644.293.446.183.013.582.418.288.512.964.757.209.870.743.934.472.061.672.081.647.165.005.618.387.786.132.070.667.312.550.890.485.505.403.716.544.778.872.192.898.136.970.375.010.274.357.427.081.353.375.089.376.782.143.678.081.963.907.852.958.319.626.245.972.043.515.519.964.702.779.613.588.349.712.822.541.044.839.786.471.769.356.045.279.301.841.071.830.155.320.088.747.687.777.679.036.500.221.131.503.499.979.504.575.800.139.618.953.991.070.532.853.066.560.299.227.796.553.133.876.492.788.471.387.931.546.473.614.682.116.573.648.894.401.928.950.676.697.741.655.042.663.160.432.049.093.844.423.771.487.564.784.140.078.212.230.826.253.199.064.718.726.442.268.863.914.621.131.705.728.934.425.922.395.324.963.886.584.520.126.767.794.172.859.617.162.078.663.860.451.891.957.154.839.984.407.506.434.112.876.487.007.685.201.545.638.383.474.365.134.030.372.400.663.301.518.149.134.174.638.959.201.138.045.043.800.404.284.330.813.643.848.642.446.512.698.685.902.193.797.912.679.197.289.529.539.539.521.449.810.714.442.565.198.278.216.365.455.011.915.421.576.258.652.895.730.215.900.302.374.917.580.018.053.651.570.992.106.524.489.590.657.900.587.570.625.744.379.186.519.421.488.520.877.903.999.425.363.706.741.239.465.560.275.250.037.105.968.752.819.708.824.658.441.551.336.227.303.560.387.773.930.367.130.490.497.737.191.158.528.370.720.480.058.629.520.189.288.387.537.450.696.844.122.530.346.385.463.324.853.957.538.538.851.116.972.967.092.999.704.879.118.387.132.854.330.328.107.827.693.937.467.287.347.538.274.594.902.731.750.244.394.464.490.197.158.501.231.457.149.736.399.183.870.541.715.899.012.829.219.207.981.459.949.070.532.694.275.286.603.818.617.396.937.996.420.009.899.103.467.474.958.498.036.633.243.200.853.183.286.600.971.353.965.870.623.674.945.011.038.720.236.579.523.799.590.804.766.006.303.206.878.480.096.490.186.884.063.849.257.660.694.003.423.814.016.730.899.173.658.625.753.071.641.084.366.116.314.733.298.531.659.941.798.174.164.443.634.526.932.808.426.209.736.448.529.423.480.715.074.538.236.051.281.356.370.205.557.298.537.804.473.520.654.085.821.963.484.446.213.447.093.493.478.105.436.390.068.371.505.767.925.883.351.873.517.238.268.285.535.258.153.641.140.173.662.913.456.752.338.996.439.694.947.459.295.909.147.107.537.308.689.499.566.226.198.769.034.114.271.441.514.924.367.202.736.600.421.291.624.626.185.612.691.833.660.991.367.335.133.933.654.955.867.668.171.632.815.842.871.816.078.544.099.135.459.253.981.792.139.261.130.052.057.679.273.911.020.068.865.592.888.176.034.404.772.670.409.620.548.469.870.812.571.242.224.766.785.903.597.368.375.801.740.705.093.657.243.994.581.280.287.940.391.862.903.474.036.903.047.520.272.297.974.628.936.388.853.144.078.848.752.869.911.250.165.499.502.068.388.850.141.263.761.703.550.768.959.086.097.423.353.432.540.696.692.709.809.622.981.960.562.366.641.400.485.907.192.259.570.580.297.159.789.983.018.699.762.191.288.261.917.134.227.355.803.146.407.182.277.730.848.943.505.098.284.684.041.046.525.667.958.811.928.772.384.811.052.101.237.341.231.033.701.944.375.149.189.914.793.181.092.191.086.396.866.352.334.748.449.769.231.395.635.567.602.908.214.466.139.766.841.051.481.060.326.459.152.770.437.486.389.005.272.441.707.123.755.307.243.210.430.531.732.049.068.456.449.825.816.806.929.054.541.919.639.401.470.321.710.848.514.620.892.330.591.633.550.185.766.036.860.943.234.581.259.615.801.049.107.411.151.644.671.751.101.708.383.202.164.583.986.339.558.524.487.030.803.831.599.546.653.763.421.487.059.629.455.443.224.227.763.350.548.492.160.495.112.475.253.019.491.484.247.273.334.953.981.051.246.738.563.060.559.894.847.220.100.004.561.228.283.032.663.663.490.607.024.016.762.521.772.414.059.741.405.138.019.007.589.458.785.491.351.732.514.198.581.173.710.128.958.313.941.112.314.580.263.851.166.875.856.301.200.483.439.518.868.872.555.780.218.919.671.127.410.917.048.448.654.558.133.796.680.189.124.421.049.177.485.066.749.163.860.074.544.198.930.562.352.634.858.744.568.269.433.666.369.070.064.786.744.129.288.067.330.503.196.284.249.790.618.086.683.008.

30

925.426.583.395.932.006.123.930.513.722.216.691.405.542.976.461.576.026.663.694.155.384.806.250.863.078.586.676.024.022.532.674.700.237.232.099.790.022.638.523.214.039.980.502.078.812.635.646.671.328.978.030.000.908.199.777.004.485.283.855.464.685.557.392.854.298.946.097.263.278.796.029.029.961.820.226.986.513.291.912.643.982.774.325.100.751.655.317.537.416.095.015.757.565.889.337.111.322.486.974.074.849.380.906.325.640.899.329.823.738.483.073.861.429.628.431.453.876.727.125.004.662.470.132.712.133.930.125.059.635.927.596.905.880.616.217.949.752.457.825.440.734.336.306.740.027.529.307.085.515.497.399.778.432.231.317.434.153.722.079.091.833.009.196.835.600.565.804.017.627.356.689.466.438.397.325.852.365.287.526.877.981.067.745.783.317.426.861.460.181.565.818.435.176.585.563.553.263.968.772.882.565.448.961.613.374.975.233.510.610.270.510.643.855.817.864.951.332.096.384.347.681.579.804.133.304.412.738.579.769.371.697.995.736.436.783.801.260.965.236.316.121.819.658.236.938.848.772.963.092.662.966.740.383.338.170.368.504.160.178.690.538.808.181.150.207.965.829.454.043.970.734.605.092.665.203.282.180.306.998.737.108.740.832.392.018.885.366.175.510.813.263.906.162.171.005.118.202.219.769.546.450.218.933.629.837.449.315.986.873.995.014.879.164.622.028.047.281.551.856.288.025.869.366.226.377.962.913.079.159.223.152.980.479.974.082.188.868.134.816.820.741.028.507.402.946.930.810.315.225.846.318.849.761.982.037.990.066.600.742.078.985.463.760.789.106.179.824.322.081.072.404.870.576.639.155.178.597.387.895.741.078.337.970.325.011.843.427.140.903.118.912.861.728.232.683.648.500.345.589.286.390.175.032.466.675.195.084.882.765.161.130.942.389.193.657.015.081.427.997.289.345.223.523.148.674.977.428.470.938.294.436.264.427.377.992.263.874.282.907.014.732.045.997.521.204.039.060.022.970.634.867.936.562.050.371.836.147.130.070.902.356.551.551.316.667.000.212.465.890.035.317.022.959.564.724.972.037.608.536.949.462.213.866.438.499.643.572.609.436.856.201.172.162.249.098.182.377.970.815.406.162.324.050.253.705.846.134.150.989.653.354.393.374.265.505.371.966.391.919.009.566.402.477.187.481.459.454.261.412.338.219.115.016.224.152.566.657.471.830.602.849.081.762.507.889.894.286.772.740.319.563.048.102.338.426.499.339.934.988.291.650.899.581.977.816.703.568.304.505.456.412.329.805.350.562.463.820.858.908.729.713.680.280.814.714.554.636.119.382.888.658.716.766.350.039.873.582.718.858.924.117.791.327.355.333.060.161.513.935.413.057.383.024.626.266.388.104.235.168.826.332.573.188.570.849.559.945.315.731.231.480.139.024.165.079.490.103.997.398.738.515.187.430.543.249.422.093.761.887.076.959.636.534.766.976.657.766.190.468.183.508.532.583.690.193.944.749.450.489.143.056.996.156.898.388.636.551.440.241.518.490.912.419.187.703.156.009.441.384.697.748.060.593.244.163.827.938.361.962.226.484.334.594.535.539.173.840.754.798.551.270.516.667.050.496.731.065.869.507.321.920.919.270.878.338.465.956.919.689.177.567.361.370.314.710.280.030.473.826.043.672.826.789.413.194.236.074.292.601.814.069.920.257.625.928.283.832.406.785.075.494.703.228.337.816.044.689.608.816.258.562.393.204.100.895.940.889.180.970.760.107.109.380.326.213.421.017.610.476.805.064.077.669.496.648.922.573.499.327.690.724.180.178.120.388.273.960.987.627.968.879.558.583.043.095.376.705.550.199.191.021.286.378.462.589.686.405.293.702.483.227.357.466.451.253.247.376.011.216.255.490.247.352.656.747.435.617.419.406.346.634.947.543.340.002.158.660.581.492.596.821.313.105.549.952.810.516.800.577.681.304.284.116.785.753.044.174.476.706.869.021.994.353.640.048.589.565.581.924.137.759.347.425.774.607.724.917.148.319.421.422.687.770.421.201.675.030.116.537.294.507.236.323.292.313.150.944.942.750.893.948.610.209.118.785.264.533.823.896.568.610.555.711.575.854.192.944.536.266.330.714.967.100.611.360.624.738.995.194.581.459.406.070.820.894.872.744.466.524.729.935.157.031.492.175.464.468.567.870.881.582.761.271.097.372.624.220.434.628.657.735.130.065.084.550.092.388.586.761.683.200.773.265.886.960.960.452.490.687.027.022.830.049.586.277.869.226.263.630.175.237.158.843.534.292.291.003.883.133.850.812.783.183.100.571.495.765.339.152.432.314.167.957.221.029.138.565.573.829.899.302.195.290.825.

31

772.526.183.743.964.173.790.228.611.192.501.654.082.144.891.830.346.172.659.991.051.381.408.141.257.105.960.177.336.986.084.842.925.975.783.181.859.299.822.502.665.565.744.896.999.350.400.858.755.053.409.643.775.663.357.955.071.862.663.255.836.966.922.465.364.032.900.847.592.329.183.798.408.398.517.577.465.125.941.098.264.076.255.091.992.180.073.626.578.203.277.086.816.138.537.028.294.667.401.704.649.912.771.028.983.643.381.623.990.666.129.099.900.254.845.625.598.466.469.369.682.697.049.039.433.896.514.622.229.694.939.511.660.434.043.154.654.341.126.565.779.759.367.902.111.555.908.911.005.947.707.963.951.247.660.462.106.373.039.251.507.912.255.600.650.920.348.835.192.689.776.594.379.488.819.932.909.742.973.567.973.593.682.776.666.511.083.338.402.742.981.058.837.811.878.150.559.799.568.134.485.601.106.754.399.153.655.680.390.181.079.011.143.615.717.606.599.353.069.113.138.221.769.583.848.263.405.849.978.711.261.082.149.814.528.219.311.655.215.231.477.674.291.680.352.723.643.693.723.563.267.709.046.427.605.981.346.289.180.013.778.059.340.540.079.409.045.640.365.842.128.981.588.669.726.980.929.890.449.590.897.770.770.717.662.756.557.321.818.635.408.920.540.448.197.962.358.293.946.973.145.169.436.334.222.382.843.495.671.076.759.497.044.546.060.643.800.127.428.205.900.283.277.440.700.839.560.328.910.569.409.904.424.449.283.012.770.825.521.878.522.632.024.245.620.547.645.717.447.254.379.107.155.964.792.664.126.225.184.844.373.849.325.631.505.412.119.004.097.493.480.791.410.583.383.081.107.888.281.419.099.235.779.383.229.680.188.384.017.779.292.617.921.370.922.372.983.282.503.298.791.017.021.921.784.715.440.301.789.087.148.507.824.536.335.557.675.207.298.935.023.348.409.468.791.607.206.887.572.163.473.596.233.634.095.514.588.426.193.776.677.103.790.030.103.697.798.488.340.857.651.485.831.498.976.810.338.247.798.891.274.171.430.792.711.417.218.119.018.685.594.870.181.189.724.154.993.802.014.008.520.621.181.498.724.733.019.671.519.274.815.607.394.471.144.930.910.973.953.871.874.677.234.320.837.673.786.173.747.996.926.108.182.160.592.819.930.285.028.243.565.270.434.019.143.599.764.939.825.434.835.631.361.977.378.834.574.498.476.196.067.276.279.197.970.185.482.513.387.075.992.762.031.284.719.426.518.736.333.645.560.539.947.459.083.154.992.047.508.021.177.432.253.410.186.114.503.046.573.992.891.136.350.397.906.727.213.151.063.042.796.603.505.508.078.503.029.848.207.956.861.976.168.707.326.309.643.927.627.932.082.478.629.958.930.389.043.300.695.953.852.245.762.426.688.198.844.500.968.552.103.443.945.388.825.434.581.952.086.254.624.804.849.323.094.655.174.896.563.225.863.492.656.231.228.848.145.999.054.507.558.064.876.783.575.233.406.881.642.671.081.875.445.728.176.973.719.623.510.405.841.272.435.458.186.174.137.551.447.332.986.198.005.631.148.179.254.729.286.590.093.654.809.867.706.726.135.218.368.269.085.682.415.849.010.927.074.017.886.271.791.094.281.967.768.009.321.122.158.979.102.288.222.564.043.650.174.535.214.025.103.753.469.650.857.266.570.607.612.767.932.187.496.888.032.409.095.242.429.364.768.509.011.462.430.863.422.253.155.131.799.344.313.616.840.024.345.057.582.809.274.140.558.256.227.759.257.581.371.536.142.332.899.517.951.052.968.071.988.191.897.884.803.413.540.626.817.984.476.492.395.370.547.098.850.219.684.882.276.641.367.409.472.540.871.594.231.816.700.092.552.991.708.816.738.672.526.158.318.251.399.000.404.719.419.039.203.630.576.102.697.834.973.314.012.448.002.803.604.296.275.826.739.467.016.237.569.391.742.822.931.453.251.705.873.513.636.865.215.811.264.417.093.048.041.562.413.771.425.101.516.307.271.586.711.402.176.333.163.244.449.446.241.366.835.142.172.999.167.198.149.195.445.076.785.771.746.105.830.099.899.755.240.514.179.074.130.483.590.748.328.925.726.169.836.229.219.135.059.115.543.674.794.729.764.757.528.954.732.301.547.753.091.360.598.364.488.144.583.854.075.666.742.799.233.901.477.793.810.708.349.408.436.623.037.613.251.688.464.857.600.837.871.310.874.599.660.539.329.826.527.399.852.230.164.943.162.201.693.782.771.653.469.751.274.562.245.118.306.884.929.923.560.792.884.211.974.942.868.001.005.426.031.939.457.512.370.244.462.812.148.016.201.342.630.473.074.456.267.159.549.293.867.053.767.

32

659.038.056.779.949.088.426.976.254.034.728.077.254.088.135.636.860.477.186.686.146.460.626.899.902.745.613.278.153.691.400.038.339.669.537.740.544.826.562.443.805.626.535.786.384.898.572.485.701.425.917.432.386.986.347.545.005.288.812.481.364.181.536.338.248.364.278.398.726.496.016.692.123.724.237.943.548.545.828.896.728.213.046.892.088.065.007.581.346.307.309.209.881.633.136.251.506.534.455.849.280.057.342.437.035.243.462.095.729.482.589.560.053.307.406.095.527.385.109.856.353.586.250.293.946.996.179.453.801.457.868.107.800.325.660.554.532.815.889.874.967.288.747.905.456.479.104.063.078.963.446.408.392.211.991.919.068.709.916.650.366.427.193.521.435.598.108.874.243.805.361.630.829.273.485.118.140.255.826.064.152.007.353.462.362.585.456.695.582.483.253.258.528.935.582.247.912.293.014.030.259.075.488.692.057.966.145.622.502.275.707.411.579.967.571.940.743.724.826.280.576.348.051.831.482.225.559.303.226.913.313.273.222.907.950.041.784.172.300.547.498.749.060.305.878.597.291.391.606.248.972.363.355.290.948.450.836.366.255.126.157.308.403.661.054.075.454.235.785.998.143.321.284.840.030.323.563.674.035.582.184.135.793.969.720.132.762.184.242.168.341.237.392.013.032.814.153.508.891.929.231.249.031.814.578.555.987.148.412.261.004.467.680.021.274.080.915.281.269.245.527.358.835.933.371.407.227.532.308.886.439.039.983.468.705.220.620.776.921.285.207.515.612.726.428.375.426.237.913.651.622.946.234.427.409.596.825.985.581.043.754.174.115.704.264.121.004.641.271.992.679.265.413.033.893.641.001.624.386.885.626.557.671.648.182.297.944.501.373.983.864.934.057.502.341.197.478.541.199.763.789.436.638.185.843.621.570.691.204.086.149.963.373.213.056.836.154.420.869.373.597.108.314.030.426.861.451.170.412.906.079.361.626.709.359.071.741.472.975.878.756.611.036.628.962.564.131.510.773.383.998.510.774.234.687.088.992.540.536.801.810.312.009.857.409.980.694.442.074.071.368.789.046.025.521.655.489.144.964.103.881.538.946.439.359.229.988.944.312.506.134.360.805.262.745.863.808.061.869.660.883.874.870.254.525.608.385.737.401.649.750.056.276.841.658.403.694.887.805.329.366.370.830.594.546.417.565.156.055.684.539.618.472.323.821.222.897.413.937.507.741.026.226.089.596.583.549.804.469.233.405.511.618.762.392.692.059.158.528.940.278.631.113.978.465.479.265.717.747.579.601.226.777.431.147.475.637.428.062.537.631.976.626.723.681.685.522.162.724.527.759.014.616.906.930.247.421.641.947.237.717.892.852.919.071.511.473.334.229.604.683.532.898.578.628.709.926.886.572.101.867.418.845.888.405.030.606.641.737.680.827.364.610.079.316.942.421.258.233.285.125.320.063.076.534.522.312.971.471.190.655.386.105.932.096.367.520.442.927.148.501.744.391.319.945.868.134.556.799.101.897.247.505.920.799.352.053.517.687.463.088.862.122.452.649.397.230.334.908.938.410.526.023.688.356.915.746.631.269.692.332.001.052.996.469.934.456.965.145.092.274.742.595.090.221.896.958.122.113.128.807.070.503.152.302.237.929.958.117.399.858.228.931.530.449.344.428.061.096.653.254.595.757.996.596.012.868.178.203.613.096.503.557.726.524.712.340.939.850.503.491.212.011.915.661.283.110.007.772.667.503.178.059.477.499.819.303.182.705.335.332.834.624.459.158.622.179.543.934.574.194.844.832.941.235.410.788.824.896.970.294.171.227.029.654.839.529.716.920.953.122.839.133.725.722.885.792.046.979.734.701.368.899.651.860.091.368.672.457.694.068.705.142.153.704.665.956.590.116.443.066.776.198.119.837.317.198.038.690.756.679.423.778.649.983.031.717.953.241.414.565.639.804.415.349.200.132.200.388.959.885.387.334.859.365.611.381.421.757.473.594.124.140.300.845.948.165.657.326.321.402.045.526.457.877.175.261.783.902.945.741.916.463.141.522.561.420.203.700.933.323.680.476.307.717.876.485.388.538.676.825.388.861.848.354.773.776.158.304.137.581.932.840.046.496.698.754.994.538.681.513.128.446.619.105.733.675.593.860.631.122.275.044.806.365.964.738.480.695.283.198.893.795.471.985.032.076.312.138.217.279.280.685.956.908.551.225.895.651.949.636.997.717.323.872.407.213.019.526.947.214.106.014.685.651.896.720.810.620.041.619.457.350.403.927.319.816.530.507.040.458.205.468.686.881.230.845.683.976.768.482.406.242.102.360.072.859.068.637.840.237.080.897.680.414.672.874.457.279.137.290.351.731.285.361.

33

826.730.747.132.309.723.457.291.984.363.054.662.169.783.469.965.067.810.677.272.716.989.229.549.869.544.997.643.420.048.951.962.117.330.583.406.227.256.546.424.931.564.492.817.317.988.500.941.141.515.774.213.641.259.043.678.844.901.192.292.384.108.515.150.131.680.045.281.832.808.388.783.271.580.422.344.334.093.295.411.984.846.535.779.091.842.904.018.189.972.599.953.261.108.377.814.593.515.723.506.074.528.068.379.258.018.331.670.636.324.449.650.928.795.940.284.933.846.400.055.459.917.356.112.116.335.784.979.086.949.945.956.369.581.741.604.812.238.482.252.014.218.364.035.269.595.199.869.272.652.197.509.591.599.816.699.129.577.988.601.552.450.772.600.803.111.730.257.689.434.740.396.645.678.128.683.234.080.110.444.791.146.574.502.596.759.538.726.267.567.009.665.497.064.869.837.428.440.984.372.682.405.795.705.294.960.282.556.025.175.630.442.242.188.965.858.646.033.509.764.922.752.662.123.652.688.977.422.399.818.270.845.289.060.360.558.386.646.605.078.985.503.263.150.878.101.365.939.532.848.349.218.710.579.326.324.704.979.570.885.404.956.464.093.057.902.572.550.952.611.083.546.856.831.967.686.482.879.514.544.641.366.849.189.646.088.116.922.576.717.467.199.925.666.186.734.857.012.684.711.404.619.122.222.064.933.395.604.978.601.744.611.743.951.113.455.953.556.328.610.156.977.263.958.739.929.187.328.674.786.548.136.811.449.687.282.949.959.464.723.918.992.516.804.733.474.011.918.958.194.431.485.725.908.061.897.509.325.645.763.342.150.537.353.562.047.224.435.251.469.859.024.539.544.189.504.439.300.849.183.351.131.681.465.155.500.713.348.848.128.795.541.773.814.600.838.142.333.161.455.411.590.379.382.669.821.711.558.360.682.069.527.214.815.090.496.728.243.251.582.786.643.214.565.620.855.793.063.391.148.893.746.148.386.079.869.990.725.018.691.105.346.934.009.990.519.445.508.612.266.631.423.508.661.251.265.674.502.497.598.756.950.398.610.432.628.407.748.730.127.550.466.119.774.356.165.205.370.842.865.986.184.667.867.735.553.498.989.258.556.008.873.856.154.427.543.324.162.434.290.854.221.385.621.825.103.918.209.962.411.496.521.118.503.100.393.801.218.804.498.094.335.898.100.551.174.651.127.818.634.061.508.921.345.797.071.094.228.891.630.661.197.651.503.176.297.804.083.407.503.387.958.738.779.823.231.728.054.874.323.709.008.933.134.448.624.438.426.846.944.206.222.892.217.386.031.225.046.088.111.368.006.652.541.888.421.175.452.254.177.582.653.499.369.929.651.138.600.800.741.027.340.620.230.487.986.645.086.949.202.737.931.216.665.994.821.930.610.870.313.709.126.952.399.737.132.540.385.309.975.814.913.154.995.423.990.139.861.558.072.123.394.602.975.459.760.316.483.377.007.587.771.825.828.734.693.080.155.411.845.165.392.580.769.408.122.778.403.721.414.643.069.282.531.176.939.619.244.341.314.801.506.617.973.949.652.070.744.757.191.107.303.628.412.599.066.986.095.533.824.720.205.929.364.949.877.803.945.229.268.870.501.245.351.146.940.626.053.304.178.852.435.521.047.034.575.113.667.734.617.754.500.661.963.818.241.921.208.808.397.380.702.506.120.671.386.228.596.514.804.976.192.096.364.433.057.894.515.026.542.453.389.519.706.976.703.667.448.822.190.555.676.718.700.117.911.969.937.613.026.278.777.664.685.898.739.837.320.198.293.385.475.856.981.545.160.467.595.781.051.059.780.865.926.310.684.871.742.256.366.073.970.371.133.315.571.897.122.174.711.575.709.464.736.881.167.901.286.125.630.981.315.652.760.228.531.501.360.562.481.714.702.756.523.436.424.625.827.221.912.993.808.147.277.963.411.976.521.259.544.076.749.554.395.312.420.787.932.593.342.283.316.285.195.016.299.514.530.112.793.605.661.009.694.234.819.494.848.120.085.316.782.419.587.272.850.426.169.597.578.713.913.838.119.633.794.226.240.972.136.103.757.985.813.472.492.983.733.417.625.435.292.871.433.706.976.344.722.792.941.015.783.853.475.353.343.652.708.407.894.358.944.520.772.289.898.872.674.742.740.484.848.339.023.165.449.701.135.009.722.248.795.133.268.731.580.400.777.616.995.446.938.112.267.988.973.386.493.287.027.448.125.279.191.613.549.280.379.125.833.943.633.785.510.826.843.274.216.751.522.821.192.213.364.520.504.701.441.736.524.853.373.673.052.775.993.304.466.331.746.228.693.662.454.820.127.855.966.813.344.427.298.724.428.725.099.997.883.815.935.895.069.759.266.742.

34

277.754.452.785.223.198.225.395.420.126.605.324.413.921.784.656.592.350.591.352.564.504.974.300.115.154.753.203.724.700.562.601.048.897.858.864.627.332.306.107.639.009.429.847.348.705.213.685.734.646.779.666.270.739.036.757.264.798.706.366.717.082.817.778.569.182.345.581.632.626.889.029.804.301.732.837.440.846.709.600.095.468.860.501.877.676.107.363.490.475.302.176.894.215.736.152.697.335.549.229.452.050.423.251.959.642.216.005.712.351.909.883.737.110.650.640.462.571.917.763.261.363.644.073.123.974.043.524.360.604.898.099.425.172.889.556.529.695.558.739.041.335.141.026.988.248.048.451.686.410.941.948.124.137.851.414.856.210.522.426.849.851.233.491.932.732.885.967.027.530.009.056.057.402.771.477.463.563.238.426.347.491.158.563.391.282.193.749.698.317.202.444.556.006.834.819.749.453.954.930.404.948.709.142.513.769.790.815.924.782.193.134.271.436.422.848.089.193.949.751.296.635.731.423.357.305.077.059.680.607.286.683.946.238.768.086.916.583.204.287.837.660.863.511.834.993.728.933.593.808.937.204.730.088.504.695.966.202.855.914.649.160.950.495.261.084.680.473.425.111.737.346.349.089.985.911.777.231.516.870.873.939.177.555.617.687.920.936.211.839.656.465.933.603.741.490.983.524.784.059.822.082.904.379.182.908.646.050.311.226.555.221.525.131.353.718.590.197.929.076.527.523.296.879.814.974.636.095.717.173.139.423.128.292.926.745.910.753.622.348.037.564.706.386.621.357.860.725.673.799.893.777.383.372.393.100.111.289.659.123.976.786.185.291.639.121.811.380.109.387.605.587.417.307.703.776.842.374.480.742.015.933.414.250.423.785.034.404.998.958.032.376.121.677.351.175.947.202.529.445.445.936.401.420.825.684.528.930.433.252.944.659.624.845.238.478.412.682.964.290.794.197.831.339.041.628.289.280.961.253.295.881.492.994.833.187.677.795.657.247.184.627.306.823.031.754.751.592.427.656.240.090.519.326.492.351.411.153.855.659.203.497.653.462.770.140.468.613.285.037.394.262.078.538.365.518.225.127.559.150.816.567.130.846.249.692.140.367.467.726.431.422.359.675.640.846.252.299.768.341.260.198.868.569.774.664.974.070.091.355.790.173.618.477.171.135.158.031.965.745.221.288.778.483.338.621.173.773.028.791.022.498.552.695.770.080.528.271.231.903.603.215.854.033.182.673.677.732.390.815.788.923.291.201.535.547.863.632.447.795.360.680.745.410.104.801.549.274.631.814.955.304.414.156.023.001.230.861.971.851.434.410.767.058.331.813.577.376.627.313.754.621.707.570.235.149.793.939.169.583.495.718.558.280.422.601.398.926.136.081.916.186.955.960.444.642.957.309.945.276.924.310.364.925.382.102.688.860.755.193.232.624.742.167.899.507.748.620.875.816.486.118.194.585.172.223.522.356.546.049.903.369.154.893.081.468.084.918.202.273.499.880.423.009.423.734.442.043.442.694.934.267.899.815.854.723.901.445.461.547.805.887.439.744.897.656.866.467.087.013.666.776.411.222.956.145.372.700.199.126.267.175.839.336.354.697.272.954.222.295.918.286.846.413.156.648.851.519.834.374.935.146.754.481.915.258.661.178.319.472.647.102.971.415.994.832.283.713.206.319.621.947.821.269.223.645.603.105.450.532.514.841.665.948.015.197.296.250.963.551.144.435.401.390.468.397.855.102.373.334.213.221.118.521.208.171.197.843.587.891.059.348.455.545.037.263.508.688.267.131.949.763.073.780.308.342.035.220.698.921.760.014.114.069.597.372.773.230.214.821.135.322.470.353.203.957.525.649.770.669.877.565.941.375.280.100.373.163.475.862.528.264.915.280.311.167.525.218.394.152.080.634.184.302.889.791.962.473.579.970.683.141.581.471.367.167.366.398.823.859.033.794.514.373.537.111.820.368.570.242.834.580.684.092.034.921.486.224.811.443.675.327.825.350.442.872.215.580.087.814.400.790.012.284.032.893.430.901.019.879.204.063.101.926.784.770.326.535.783.846.811.529.051.794.156.407.477.187.456.890.340.466.432.830.589.923.960.026.722.742.149.184.270.686.318.060.077.977.380.394.275.787.685.722.401.626.576.051.886.422.849.376.510.886.640.359.204.410.603.860.752.477.917.290.151.466.704.245.358.220.930.544.452.648.689.125.784.308.911.759.593.267.785.772.940.089.830.329.201.292.694.811.210.391.594.173.316.339.596.562.858.769.040.282.000.587.564.934.571.203.658.346.546.405.901.077.095.655.346.497.964.058.797.249.584.413.213.202.466.300.584.026.658.522.430.836.645.691.796.402.536.511.023.785.188.

35

368.038.263.793.065.285.747.473.834.067.395.416.528.863.125.144.331.209.051.730.445.172.788.920.025.966.144.945.622.750.727.071.769.079.507.716.188.347.244.949.699.706.258.134.052.274.553.137.262.087.312.709.718.464.178.196.050.537.438.923.956.944.269.088.448.615.730.140.394.291.854.648.304.518.460.320.125.944.437.200.877.090.952.384.451.980.339.029.248.077.677.303.351.618.474.563.787.853.633.449.012.806.063.571.691.919.714.073.305.893.533.618.678.943.338.276.923.320.031.837.250.014.518.352.089.679.990.940.060.604.580.188.868.126.181.823.515.089.965.208.831.121.909.774.681.600.755.207.102.403.082.769.222.517.881.229.388.340.462.909.419.478.821.493.296.594.574.233.267.938.694.393.658.576.591.717.632.521.049.235.386.262.891.417.365.005.921.683.706.006.284.478.099.570.816.690.996.105.865.157.573.451.865.316.909.883.613.381.642.675.289.627.644.828.615.531.318.743.459.872.161.460.363.137.442.976.038.994.898.964.975.630.432.955.663.939.378.214.779.188.148.425.633.156.064.943.513.557.006.413.191.010.833.856.272.339.131.999.107.769.951.621.083.465.515.007= 149356 = 28 = 1

(En un reloj de 18 horas las superposiciones de los números primos son correctas)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181 2 3 5 7 11 13 1719 23 29 3137 41 43 47 53…

(31 = 3 + 1 = 4 = 1 + 3 = 4) (53 = 5 + 3 = 8 = 1 + 7 = 8)

36

(En relojes de otras horas las superposiciones de los números primos son incorrectas: (24, 29…)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 241 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53…

(31 = 3 + 1 = 4 ≠ 7) (53 = 5 + 3 = 8 ≠ 5)

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

37

1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53…

(31 = 3 + 1 = 4 ≠ 2) (53 = 5 + 3 = 8 ≠ 24 ≠ 6)

i) y este análisis es aplicable a todos los números primos conocidos y por conocer.

j) veamos como final la descomposición horaria del reloj de 18 horas en los primeros 1000 números y el modelo de borrado de los números compuestos:

(Hora 1) 1 = (NUMERO PRIMO) (Giro 1º)

(Hora 5) 5 = (NUMERO PRIMO)  (Hora 7) 7 = (NUMERO PRIMO) 

(Hora 11) 11 = (NUMERO PRIMO) 

(Hora 13) 13 = (NUMERO PRIMO) 

(Hora 17) 17 = (NUMERO PRIMO)

(Hora 1) 19 = (NUMERO PRIMO) (Giro 2º)

(Hora 5) 23 = (NUMERO PRIMO) 

(Hora 7) 25 = (5x5)

(Hora 11) 29 = (NUMERO PRIMO) 

(Hora 13) 31 = (NUMERO PRIMO) 

(Hora 17) 35 = (5x7)

(Hora 1) 37 = (NUMERO PRIMO) (Giro 3º) 

(Hora 5) 41 = (NUMERO PRIMO) 

(Hora 7) 43 = (NUMERO PRIMO) 

(Hora 11) 47 = (NUMERO PRIMO) 

(Hora 13) 49 = (7x7)

(Hora 17) 53 = (NUMERO PRIMO)

38

 

(Hora 1) 55 = (5x11) (Giro 4º) 

(Hora 5) 59 = (NUMERO PRIMO) (Hora 7) 61 = (NUMERO PRIMO) 

(Hora 11) 65 = (5x13)

(Hora 13) 67 = (NUMERO PRIMO) 

(Hora 17) 71 = (NUMERO PRIMO) 

(Hora 1) 73 = (NUMERO PRIMO) (Giro 5º) 

(Hora 5) 77 = (7x11)

(Hora 7) 79 = (NUMERO PRIMO) 

(Hora 11) 83 = (NUMERO PRIMO) 

(Hora 13) 85 = (5x17)

(Hora 17) 89 = (NUMERO PRIMO) 

(Hora 1) 91 = (7x13) (Giro 6º) 

(Hora 5) 95 = (5x19)

(Hora 7) 97 = (NUMERO PRIMO) 

(Hora 11) 101 = (NUMERO PRIMO) 

(Hora 13) 103 = (NUMERO PRIMO) (Hora 17) 107 = (NUMERO PRIMO)  (Hora 1) 109 = (NUMERO PRIMO) (Giro 7º) 

(Hora 5) 113 = (NUMERO PRIMO) 

(Hora 7) 115 = (5x23)

(Hora 11) 119 = (7x17)

(Hora 13) 121 = (11x11)

(Hora 17) 125 = (5x25)

(Hora 1) 127 = (NUMERO PRIMO) (Giro 8º) 

39

 (Hora 5) 131 = (NUMERO PRIMO) 

(Hora 7) 133 = (7x19)

(Hora 11) 137 = (NUMERO PRIMO) 

(Hora 13) 139 = (NUMERO PRIMO) 

(Hora 17) 143 = (11x13)

(Hora 1) 145 = (5x29) (Giro 9º) 

(Hora 5) 149 = (NUMERO PRIMO) 

(Hora 7) 151 = (NUMERO PRIMO) 

(Hora 11) 155 = (5x31)

(Hora 13) 157 = (NUMERO PRIMO) 

(Hora 17) 161 = (7x23)

(Hora 1) 163 = (NUMERO PRIMO) (Giro 10º) 

(Hora 5) 167 = (NUMERO PRIMO) 

(Hora 7) 169 = (13x13) (Hora 11) 173 = (NUMERO PRIMO) 

(Hora 13) 175 = (5x35) (7x25)

(Hora 17) 179 = (NUMERO PRIMO) 

(Hora 1) 181 = (NUMERO PRIMO) (Giro 11º) 

(Hora 5) 185 = (5x37) (Hora 7) 187 = (11x17)

(Hora 11) 191 = (NUMERO PRIMO) (Hora 13) 193 = (NUMERO PRIMO) 

(Hora 17) 197 = (NUMERO PRIMO) 

(Hora 1) 199 = (NUMERO PRIMO) (Giro 12º) 

40

(Hora 5) 203 = (7x29) (Hora 7) 205 = (5x41)

(Hora 11) 209 = (11x19)

(Hora 13) 211 = (NUMERO PRIMO)  (Hora 17) 215 = (5x43)

(Hora 1) 217 = (7x31) (Giro 13º) 

(Hora 5) 221 = (13x17)

(Hora 7) 223 = (NUMERO PRIMO)  (Hora 11) 227 = (NUMERO PRIMO) (Hora 13) 229 = (NUMERO PRIMO) 

(Hora 17) 233 = (NUMERO PRIMO) 

(Hora 1) 235 = (5x47) (Giro 14º) 

(Hora 5) 239 = (NUMERO PRIMO) (Hora 7) 241 = (NUMERO PRIMO) 

(Hora 11) 245 = (5x49)

(Hora 13) 247 = (13x19)

(Hora 17) 251 = (NUMERO PRIMO) 

(Hora 1) 253 = (11x23) (Giro 15º) 

(Hora 5) 257 = (NUMERO PRIMO) (Hora 7) 259 = (7x37)

(Hora 11) 263 = (NUMERO PRIMO) (Hora 13) 265 = (5x53)

(Hora 17) 269 = (NUMERO PRIMO) 

(Hora 1) 271 = (NUMERO PRIMO) (Giro 16º) 

41

 (Hora 5) 275 = (5x55)

(Hora 7) 277 = (NUMERO PRIMO) 

(Hora 11) 281 = (NUMERO PRIMO) 

(Hora 13) 283 = (NUMERO PRIMO) 

(Hora 17) 287 = (7x41)

(Hora 1) 289 = (17x17) (Giro 17º) 

(Hora 5) 293 = (NUMERO PRIMO) 

(Hora 7) 295 = (5x59)

(Hora 11) 299 = (13x23)

(Hora 13) 301 = (7x43)

(Hora 17) 305 = (5x61)

(Hora 1) 307 = (NUMERO PRIMO) (Giro 18º) 

(Hora 5) 311 = (NUMERO PRIMO) 

(Hora 7) 313 = (NUMERO PRIMO) 

(Hora 11) 317 = (NUMERO PRIMO) 

(Hora 13) 319 = (11x 29)

(Hora 17) 323 = (17x19)

(Hora 1) 325 = (5x 65) (Giro 19º) 

(Hora 5) 329 = (7x 47)

(Hora 7) 331 = (NUMERO PRIMO) 

(Hora 11) 335 = (5x67)

(Hora 13) 337 = (NUMERO PRIMO) 

(Hora 17) 341 = (11x31)

(Hora 1) 343 = (7x 49) (Giro 20º) 

(Hora 5) 347 = (NUMERO PRIMO)

42

(Hora 7) 349 = (NUMERO PRIMO) 

(Hora 11) 353 = (NUMERO PRIMO)  (Hora 13) 355 = (5x71)

(Hora 17) 359 = (NUMERO PRIMO) 

(Hora 1) 361 = (19x19) (Giro 21º) 

(Hora 5) 365 = (5x73)

(Hora 7) 367 = (NUMERO PRIMO) 

(Hora 11) 371 = (7x53) (Hora 13) 373 = (NUMERO PRIMO) 

(Hora 17) 377 = (13x29)

(Hora 1) 379 = (NUMERO PRIMO) (Giro 22º)  

(Hora 5) 383 = (NUMERO PRIMO) 

(Hora 7) 385 = (5x77) (Hora 11) 389 = (NUMERO PRIMO)

 (Hora 13) 391 = (17x23)

(Hora 17) 395 = (5x79)

(Hora 1) 397 = (NUMERO PRIMO) (Giro 23º) 

(Hora 5) 401 = (NUMERO PRIMO) (Hora 7) 403 = (13x31)

(Hora 11) 407 = (11x37)

(Hora 13) 409 = (NUMERO PRIMO) 

(Hora 17) 413 = (7x59)

(Hora 1) 415 = (5x83) (Giro 24º) 

(Hora 5) 419 = (NUMERO PRIMO) 

43

(Hora 7) 421 = (NUMERO PRIMO) 

(Hora 11) 425 = (5x85)

(Hora 13) 427 = (7x61)

(Hora 17) 431 = (NUMERO PRIMO) 

(Hora 1) 433 = (NUMERO PRIMO) (Giro 25º) 

(Hora 5) 437 = (19x23)

(Hora 7) 439 = (NUMERO PRIMO)

(Hora 11) 443 = (NUMERO PRIMO) 

(Hora 13) 445 = (5x 89)

(Hora 17) 449 = (NUMERO PRIMO)

 (Hora 1) 451 = (11x41) (Giro 26º) 

(Hora 5) 455 = (5x91)

(Hora 7) 457 = (NUMERO PRIMO) 

(Hora 11) 461 = (NUMERO PRIMO) 

(Hora 13) 463 = (NUMERO PRIMO) 

(Hora 17) 467 = (NUMERO PRIMO) 

(Hora 1) 469 = (7x67) (Giro 27º) 

(Hora 5) 473 = (11x43)

(Hora 7) 475 = (5x95)

(Hora 11) 479 = (NUMERO PRIMO) 

(Hora 13) 481 = (13x37)

(Hora 17) 485 = (5x 97)

(Hora 1) 487 = (NUMERO PRIMO) (Giro 28º) 

(Hora 5) 491 = (NUMERO PRIMO) 

(Hora 7) 493 = (17x29)

44

(Hora 11) 497 = (7x71)

(Hora 13) 499 = (NUMERO PRIMO) 

(Hora 17) 503 = (NUMERO PRIMO) 

(Hora 1) 505 = (5x101) (Giro 29º) 

(Hora 5) 509 = (NUMERO PRIMO) 

(Hora 7) 511 = (7x73)

(Hora 11) 515 = (5x103)

(Hora 13) 517 = (11x 47)

(Hora 17) 521 = (NUMERO PRIMO) 

(Hora 1) 523 = (NUMERO PRIMO) (Giro 30º) 

(Hora 5) 527 = (17x31)

(Hora 7) 529 = (23x23)

(Hora 11) 533 = (13x41)

(Hora 13) 535 = (5x107)

(Hora 17) 539 = (7x77)

(Hora 1) 541 = (NUMERO PRIMO) (Giro 31º)  (Hora 5) 545 = (5x109)

(Hora 7) 547 = (NUMERO PRIMO) 

(Hora 11) 551 = (19x29)

(Hora 13) 553 = (7x79)

(Hora 17) 557 = (NUMERO PRIMO) 

(Hora 1) 559 = (13x43) (Giro 32º) 

(Hora 5) 563 = (NUMERO PRIMO) 

(Hora 7) 565 = (5x113)

(Hora 11) 569 = (NUMERO PRIMO) 

(Hora 13) 571 = (NUMERO PRIMO)

45

 (Hora 17) 575 = (5x115)

(Hora 1) 577 = (NUMERO PRIMO) (Giro 33º)  (Hora 5) 581 = (7x83)

(Hora 7) 583 = (11x 53) (Hora 11) 587 = (NUMERO PRIMO) 

(Hora 13) 589 = (19x31)

(Hora 17) 593 = (NUMERO PRIMO) 

(Hora 1) 595 = (5x119) (Giro 34º) 

(Hora 5) 599 = (NUMERO PRIMO) 

(Hora 7) 601 = (NUMERO PRIMO) 

(Hora 11) 605 = (5x121)

(Hora 13) 607 = (NUMERO PRIMO) 

(Hora 17) 611 = (13x47)

(Hora 1) 613 = (NUMERO PRIMO) (Giro 35º) 

(Hora 5) 617 = (NUMERO PRIMO) 

(Hora 7) 619 = (NUMERO PRIMO) 

(Hora 11) 623 = (7x89)

(Hora 13) 625 = (5x125)

(Hora 17) 629 = (17x37)

(Hora 1) 631 = (NUMERO PRIMO) (Giro 36º)  (Hora 5) 635 = (5x127)

(Hora 7) 637 = (7x91)

(Hora 11) 641 = (NUMERO PRIMO) 

(Hora 13) 643 = (NUMERO PRIMO) 

(Hora 17) 647 = (NUMERO PRIMO) 

46

(Hora 1) 649 = (11x59) (Giro 37º) 

(Hora 5) 653 = (NUMERO PRIMO) 

(Hora 7) 655 = (5x131)

(Hora 11) 659 = (NUMERO PRIMO) 

(Hora 13) 661 = (NUMERO PRIMO) 

(Hora 17) 665 = (5x133)

(Hora 1) 667 = (23x29) (Giro 38º) 

(Hora 5) 671 = (11x61)

(Hora 7) 673 = (NUMERO PRIMO) 

(Hora 11) 677 = (NUMERO PRIMO) 

(Hora 13) 679 = (7x97)

(Hora 17) 683 = (NUMERO PRIMO) 

(Hora 1) 685 = (5x137) (Giro 39º) 

(Hora 5) 689 = (13x53)

(Hora 7) 691 = (NUMERO PRIMO) 

(Hora 11) 695 = (5x139)

(Hora 13) 697 = (17x41)

(Hora 17) 701 = (NUMERO PRIMO)

 (Hora 1) 703 = (19x37) (Giro 40º) 

(Hora 5) 707 = (7x101)

(Hora 7) 709 = (NUMERO PRIMO) 

(Hora 11) 713 = (23x31)

(Hora 13) 715 = (5x143)

(Hora 17) 719 = (NUMERO PRIMO)

47

 

(Hora 1) 721 = (7x103) (Giro 41º) 

(Hora 5) 725 = (5x145)

(Hora 7) 727 = (NUMERO PRIMO) (Hora 11) 731 = (17x43)

(Hora 13) 733 = (NUMERO PRIMO) 

(Hora 17) 737 = (11x67)

(Hora 1) 739 = (NUMERO PRIMO) (Giro 42º)

(Hora 5) 743 = (NUMERO PRIMO) 

(Hora 7) 745 = (5x149)

(Hora 11) 749 = (7x107) 

(Hora 13) 751 = (NUMERO PRIMO) 

(Hora 17) 755 = (5x151)

(Hora 1) 757 = (NUMERO PRIMO) (Giro 43º)

(Hora 5) 761 = (NUMERO PRIMO) 

(Hora 7) 763 = (7x109) 

(Hora 11) 767 = (13x59)

(Hora 13) 769 = (NUMERO PRIMO) 

(Hora 17) 773 = (NUMERO PRIMO)

 (Hora 1) 775 = (5x155) (Giro 44º)

(Hora 5) 779 = (19x41)

(Hora 7) 781 = (11x71)

(Hora 11) 785 = (5x157)

(Hora 13) 787 = (NUMERO PRIMO) 

(Hora 17) 791 = (7x113)

48

(Hora 1) 793 = (13x61) (Giro 45º)

(Hora 5) 797 = (NUMERO PRIMO)  (Hora 7) 799 = (17x47)

(Hora 11) 803 = (11x73)

(Hora 13) 805 = (5x161)

(Hora 17) 809 = (NUMERO PRIMO)

 (Hora 1) 811 = (NUMERO PRIMO) (Giro 46º)

(Hora 5) 815 = (5x163)

(Hora 7) 817 = (19x43)

(Hora 11) 821 = (NUMERO PRIMO) 

(Hora 13) 823 = (NUMERO PRIMO) 

(Hora 17) 827 = (NUMERO PRIMO)

(Hora 1) 829 = (NUMERO PRIMO) (Giro 47º)

(Hora 5) 833 = (7x 119)

(Hora 7) 835 = (5x167)

(Hora 11) 839 = (NUMERO PRIMO) 

(Hora 13) 841 = (29x29)

(Hora 17) 845 = (5x169)

(Hora 1) 847 = (7x121) (Giro 48º)

(Hora 5) 851 = (23x37)

(Hora 7) 853 = (NUMERO PRIMO) (Hora 11) 857 = (NUMERO PRIMO) 

(Hora 13) 859 = (NUMERO PRIMO) 

(Hora 17) 863 = (NUMERO PRIMO)

 (Hora 1) 865 = (5x173) (Giro 49º)

(Hora 5) 869 = (11x79)  

49

(Hora 7) 871 = (13x67)

(Hora 11) 875 = (5x175)(Hora 13) 877 = (NUMERO PRIMO) 

(Hora 17) 881 = (NUMERO PRIMO)

 (Hora 1) 883 = (NUMERO PRIMO) (Giro 50º) (Hora 5) 887 = (NUMERO PRIMO) 

(Hora 7) 889 = (7x127)

(Hora 11) 893 = (19x47)

(Hora 13) 895 = (5x179) (Hora 17) 899 = (29x31)

(Hora 1) 901 = (17x53) (Giro 51º)

(Hora 5) 905 = (5x181)

(Hora 7) 907 = (NUMERO PRIMO) 

(Hora 11) 911 = (NUMERO PRIMO) (Hora 13) 913 = (11x83)

(Hora 17) 917 = (7x 131)

(Hora 1) 919 = (NUMERO PRIMO) (Giro 52º)

(Hora 5) 923 = (13x71)

(Hora 7) 925 = (5x185)

(Hora 11) 929 = (NUMERO PRIMO) 

(Hora 13) 931 = (7x133)

(Hora 17) 935 = (5x187)

 (Hora 1) 937 = (NUMERO PRIMO) (Giro 53º)

(Hora 5) 941 = (NUMERO PRIMO) 

(Hora 7) 943 = (23x41)

(Hora 11) 947 = (NUMERO PRIMO) 

50

(Hora 13) 949 = (13x73)

(Hora 17) 953 = (NUMERO PRIMO)

(Hora 1) 955 = (5x191) (Giro 54º)

(Hora 5) 959 = (7x137)

(Hora 7) 961 = (31x31)

(Hora 11) 965 = (5x193)

(Hora 13) 967 = (NUMERO PRIMO) 

(Hora 17) 971 = (NUMERO PRIMO)

(Hora 1) 973 = (7x139) (Giro 55º)

(Hora 5) 977 = (NUMERO PRIMO) 

(Hora 7) 979 = (11x89)

(Hora 11) 983 = (NUMERO PRIMO)  (Hora 13) 985 = (5x197)

(Hora 17) 989 = (23x43)

 (Hora 1) 991 = (NUMERO PRIMO) (Giro 56º)

(Hora 5) 995 = (5x199)

(Hora 7) 997 = (NUMERO PRIMO) 

51

52

53

54

55

56

57

58

59

60

61

CONCLUSION

"Hay dos hechos en torno a la distribución de los números primos que espero crean tan abrumadoramente, que quedarán por siempre grabadas en sus corazones. La primera es que a pesar de su sencilla definición y de su papel como ladrillos que construyen los números naturales, los números primos crecen como  la mala hierba alrededor de los números naturales, simulando no obedecer otra ley que la del azar, y nadie puede predecir donde brotará el siguiente. El segundo hecho es aún más asombroso ya que dice justamente lo contrario: que los números primos muestran una regularidad pasmosa, que hay leyes que gobiernan su comportamiento, y que obedecen estas leyes con precisión casi militar". (Don Zagier) (1965).

"Está claro que la segunda parte del escrito de Don Zagier, describe perfectamente el comportamiento de los números primos, y a través de la aplicación del Reloj de Sergio Aschero, esto se demuestra con absoluta claridad".

© Sergio Aschero (Doctor en Musicología) sergioaschero@gmail.com Avenida Roca 1669 (1870) Avellaneda / Provincia de Buenos Aires / Argentina

http://ascheropus.blogspot.com

62