Números Primos _el reloj de Sergio Aschero_

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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

Transcript of Números Primos _el reloj de Sergio Aschero_

Page 1: Números Primos _el reloj de Sergio Aschero_

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:

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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.

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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.

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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)

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(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

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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)

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(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) 

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 (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) 

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(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) 

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 (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)

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(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) 

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(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)

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(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)

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 (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) 

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(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)

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(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)

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(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)  

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(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) 

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(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) 

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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) [email protected] Avenida Roca 1669 (1870) Avellaneda / Provincia de Buenos Aires / Argentina

http://ascheropus.blogspot.com

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