LA CRIBA DE FIBONACCI PARA NÚMEROS PRIMOS

Hace aproximadamente un año que me percaté de una relación que existía entre la Sucesión del orden y la belleza: la Sucesión de Fibonacci  de la cual se engendra el número áureo  1.618...  y la distribución de los Números Primos.

Vamos empezando a modelar un esquema gráfico donde visualmente nos percataremos de esa  misteriosa relación:

Si hacemos una lista horizontal de los elementos de la Sucesión de Fibonacci y de manera paralela  los números naturales, podemos verificar que a partir del número primo 7  encontramos un patrón de la siguiente manera:

Los números Primos de la lista de los Naturales tienden a ser divisibles  por los elementos de la sucesión de Fibonacci en posiciones n+1 y n-1, vemos en el gráfico el trazo de diagonales y generamos la siguiente tabla para hacer un análisis:

 

 

Este patrón  para casi  10 primos nos permite intuir que se puede cumplir al infinito, pero tenemos un pequeño inconveniente, la sucesión de Fibonacci comienza a crecer considerablemente, por lo que nuestra capacidad de cálculo nos obliga  a hacer uso de un software matemático para  comprobar que la tendencia se sigue produciendo en  una cantidad considerable antes de formular toda conjetura, y es cuando hacemos uso de Wólfram Matemathica.

Comenzamos con la siguiente expresión producto de la intuición:

Si   Fibonacci(n+1) Mod  (n) =0   y   Fibonacci (n-1) Mod  (n) =0, entonces ‘n’ es un número Primo.

 

*Donde Mod representa el residuo de una división.

Encontramos el primer contraejemplo para nuestra intuición, para el caso  n=323, el cual no es un número primo, después observamos que la sucesión vuelve a retomar el patrón.

 

No todo está perdido, si bien el patrón se rompe con un par de contraejemplos,  nuestro modelo gráfico bautizado como la Criba de Fibonacci  podría encontrar  sin problema los primeros 66 números primos:  

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97, 101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191, 193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283, 293,307,311,313,317…

   Al observar que el patrón vuelve a recobrar sentido, podemos sacar provecho de este partido y generar fórmulas que produzcan una cantidad de números primos considerables, ese será nuestro punto de partida en este momento, nos fijamos una primer meta: Conseguir una expresión matemática que genere  por lo menos a los primeros 100 números primos haciendo uso del patrón observado de la sucesión de Fibonacci.

Al seguir trabajando en Wolfram Matemathica con las tendencias antes encontradas  para 700 valores de la variable ‘n’ y además de ir condicionando las fórmulas debido a contraejemplos  y parásitos encontrados, llegamos a las siguientes expresiones donde se incluyen residuos, funciones piso y  funciones techo:

QUE CORRESPONDEN A LOS PRIMEROS 125 NÚMEROS PRIMOS.

SI APLICAMOS EN WOLFRAM MATEMATHICA A LA FÓRMULA ‘PRIMOS’ LA FUNCIÓN DELETECASES PARA SUPRIMIR LOS CEROS Y SÓLO DEJAR LAS ENTIDADES PRIMALES, NOS QUEDA:

PRIMOS=A+B+C+D+E+F+G

DeleteCases [PRIMOS, 0]

 

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89, 97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173, 179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263, 269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359, 367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457, 461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569, 571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659, 661,673,677,683,691

NUESTRA FÓRMULA  PRIMOS=A+B+C+D+E+F+G    GENERA LOS PRIMEROS 125 NÚMEROS PRIMOS

NUEVA+NUEVA2+NUEVA3

EL RESULTADO:

NUEVA=

2,3,0,0,0,7,0,0,0,0,0,13,0,0,0,17,0,19,0,0,0,23,0,0,0,0,0,29, 0,31,0,0,0,0,0,37,0,0,0,41,0,43,0,0,0,47,0,0,0,0,0,53,0,0,0, 0,0,59,0,61,0,0,0,0,0,67,0,0,0,71,0,73,0,0,0,0,0,79,0,0,0,83, 0,0,0,0,0,89,0,0,0,0,0,0,0,97,0,0,0,101,0,103,0,0,0,107,0, 109,0,0,0,113,0,0,0,0,0,0,0,0,0,0,0,0,0,127,0,0,0,131,0,0, 0,0,0,137,0,139,0,0,0,0,0,0,0,0,0,149,0,151,0,0,0,0,0,157, 0,0,0,0,0,163,0,0,0,167,0,0,0,0,0,173,0,0,0,0,0,179,0,181, 0,0,0,0,0,0,0,0,0,191,0,193,0,0,0,197,0,199,0,0,0,0,0,0,0, 0,0,0,0,211,0,0,0,0,0,0,0,0,0,0,0,223,0,0,0,227,0,229,0,0, 0,233,0,0,0,0,0,239,0,241,0,0,0,0,0,0,0,0,0,251,0,0,0,0,0, 257,0,0,0,0,0,263,0,0,0,0,0,269,0,271,0,0,0,0,0,277,0,0,0, 281,0,283,0,0,0,0,0,0,0,0,0,293,0,0,0,0,0,0,0,0,0,0,0,0,0, 307,0,0,0,311,0,313,0,0,0,317,0,0,0,0,0,0,0,0,0,0,0,0,0, 331,0,0,0,0,0,337,0,0,0,0,0,0,0,0,0,347,0,349,0,0,0,353, 0,0,0,0,0,359,0,0,0,0,0,0,0,367,0,0,0,0,0,373,0,0,0,0,0, 379,0,0,0,383,0,0,0,0,0,389,0,0,0,0,0,0,0,397,0,0,0, 401,0,0,0,0,0,0,0,409,0,0,0,0,0,0,0,0,0,419,0,421,0,0,0, 0,0,0,0,0,0,431,0,433,0,0,0,0,0,439,0,0,0,443,0,0,0,0, 0,449,0,0,0,0,0,0,0,457,0,0,0,461,0,463,0,0,0,467,0,0, 0,0,0,0,0,0,0,0,0,479,0,0,0,0,0,0,0,487,0,0,0,491,0,0,0, 0,0,0,0,499,0,0,0,503,0,0,0,0,0,509,0,0,0,0,0,0,0,0,0,0, 0,521,0,523,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,541,0,0,0,0, 0,547,0,0,0,0,0,0,0,0,0,557,0,0,0,0,0,563,0,0,0,0,0,569, 0,571,0,0,0,0,0,577,0,0,0,0,0,0,0,0,0,587,0,0,0,0,0,593, 0,0,0,0,0,599,0,601,0,0,0,0,0,607,0,0,0,0,0,613,0,0,0, 617,0,619,0,0,0,0,0,0,0,0,0,0,0,631,0,0,0,0,0,0,0,0,0, 641,0,643,0,0,0,647,0,0,0,0,0,653,0,0,0,0,0,659,0,661, 0,0,0,0,0,0,0,0,0,0,0,673,0,0,0,677,0,0,0,0,0,683,0,0,0, 0,0,0,0,691,0,0,0,0,0,0,0,0,0,701,0,0,0,0,0,0,0,709,0,0, 0,0,0,0,0,0,0,719,0,0,0,0,0,0,0,727,0,0,0,0,0,733,0,0, 0,0,0,739,0,0,0,743,0,0,0,0,0,0,0,751,0,0,0,0,0,757, 0,0,0,761,0,0,0,0,0,0,0,769,0,0,0,773,0,0,0,0,0,0,0,0, 0,0,0,0,0,787,0,0,0,0,0,0,0,0,0,797,0,0,0,0,0,0,0,0, 0,0,0,809,0,811,0,0,0,0,0,0,0,0,0,821,0,823,0,0,0,827, 0,829,0,0,0,0,0,0,0,0,0,839,0,0,0,0,0,0,0,0,0,0,0, 0,0,853,0,0,0,857,0,859,0,0,0,863,0,0,0,0,0,0,0,0, 0,0,0,0,0,877,0,0,0,881,0,883,0,0,0,887,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,907,0,0,0,911,0,0,0,0,0,0,0, 919,0,0,0,0,0,0,0,0,0,929,0,0,0,0,0,0,0,937,0,0,0,941, 0,0,0,0,0,947,0,0,0,0,0,953,0,0,0,0,0,0,0,0,0,0,0,0,0,967, 0,0,0,971,0,0,0,0,0,977,0,0,0,0,0,983,0,0,0,0,0, 0,0,991,0,0,0,0,0,997,0,0,0,0,0,0,0,0,0,0,0,1009,0,0,0, 1013,0,0,0,0,0,1019,0,1021,0,0,0,0,0,0,0,0,0, 1031,0,1033,0,0,0,0,0,1039,0,0,0,0,0,0,0,0,0,1049,0, 1051,0,0,0,0,0,0,0,0,0,1061,0,1063,0,0,0,0,0,1069, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1087,0,0,0,1091,0,1093, 0,0,0,1097,0,0,0,0,0,1103,0,0,0,0,0,1109,0,0,0, 0,0,0,0,1117,0,0,0,0,0,1123,0,0,0,0,0,1129,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1151,0,1153,0,0,0,0, 0,0,0,0,0,1163,0,0,0,0,0,0,0,1171,0,0,0,0,0,0,0,0,0,1181 ,0,0,0,0,0,1187,0,0,0,0,0,1193,0,0,0,0,0,0,0,1201, 0,0,0,0,0,0,0,0,0,0,0,1213,0,0,0,1217,0,0,0,0,0,1223,0,0,0,0,0, 1229,0,1231,0,0,0,0,0,1237,0,0,0,0,0,0,0,0,0,0,0,1249, 0,0,0,0,0,0,0,0,0,1259,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 1277,0,1279,0,0,0,1283,0,0,0,0,0,1289,0,1291,0,0,0, 0,0,1297,0,0,0,1301,0,1303,0,0,0,1307,0,0,0,0,0,0,0, 0,0,0,0,1319,0,1321,0,0,0,0,0,1327,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 1361,0,0,0,0,0,1367,0,0,0,0,0,1373, 0,0,0,0,0,0,0,1381

NUEVA2=

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NUEVA3=

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EL RESULTADO FINAL:

2,3,0,5,0,7,0,0,0,11,0,13,0,0,0,17,0,19,0,0,0, 23,0,0,0,0,0,29,0,31,0,0,0,0,0,37,0,0,0,41,0, 43,0,0,0,47,0,0,0,0,0,53,0,0,0,0,0,59,0,61,0, 0,0,0,0,67,0,0,0,71,0,73,0,0,0,0,0,79,0,0,0,83, 0,0,0,0,0,89,0,0,0,0,0,0,0,97,0,0,0,101,0,103, 0,0,0,107,0,109,0,0,0,113,0,0,0,0,0,0,0,0,0,0, 0,0,0,127,0,0,0,131,0,0,0,0,0,137,0,139,0,0, 0,0,0,0,0,0,0,149,0,151,0,0,0,0,0,157,0,0,0,0,0, 163,0,0,0,167,0,0,0,0,0,173,0,0,0,0,0,179,0, 181,0,0,0,0,0,0,0,0,0,191,0,193,0,0,0,197,0, 199,0,0,0,0,0,0,0,0,0,0,0,211,0,0,0,0,0,0,0, 0,0,0,0,223,0,0,0,227,0,229,0,0,0,233,0,0, 0,0,0,239,0,241,0,0,0,0,0,0,0,0,0,251,0,0,0, 0,0,257,0,0,0,0,0,263,0,0,0,0,0,269,0,271,0,0,0,0,0,277, 0,0,0,281,0,283,0,0,0,0,0,0,0,0,0,293,0,0,0,0, 0,0,0,0,0,0,0,0,0,307,0,0,0,311,0,313,0,0,0,317, 0,0,0,0,0,0,0,0,0,0,0,0,0,331,0,0,0,0,0,337,0,0, 0,0,0,0,0,0,0,347,0,349,0,0,0,353,0,0,0,0,0,359, 0,0,0,0,0,0,0,367,0,0,0,0,0,373,0,0,0,0,0,379, 0,0,0,383,0,0,0,0,0,389,0,0,0,0,0,0,0,397,0,0,0, 401,0,0,0,0,0,0,0,409,0,0,0,0,0,0,0,0,0,419,0, 421,0,0,0,0,0,0,0,0,0,431,0,433,0,0,0,0,0,439, 0,0,0,443,0,0,0,0,0,449,0,0,0,0,0,0,0,457,0,0, 0,461,0,463,0,0,0,467,0,0,0,0,0,0,0,0,0,0,0,479, 0,0,0,0,0,0,0,487,0,0,0,491,0,0,0,0,0,0,0,499,0, 0,0,503,0,0,0,0,0,509,0,0,0,0,0,0,0,0,0,0,0,521, 0,523,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,541,0,0,0, 0,0,547,0,0,0,0,0,0,0,0,0,557,0,0,0,0,0,563, 0,0,0,0,0,569,0,571,0,0,0,0,0,577,0,0,0,0,0,0,0, 0,0,587,0,0,0,0,0,593,0,0,0,0,0,599,0,601,0, 0,0,0,0,607,0,0,0,0,0,613,0,0,0,617,0,619,0,0,0, 0,0,0,0,0,0,0,0,631,0,0,0,0,0,0,0,0,0,641,0, 643,0,0,0,647,0,0,0,0,0,653,0,0,0,0,0,659,0,661, 0,0,0,0,0,0,0,0,0,0,0,673,0,0,0,677,0,0,0,0,0, 683,0,0,0,0,0,0,0,691,0,0,0,0,0,0,0,0,0,701,0,0,0, 0,0,0,0,709,0,0,0,0,0,0,0,0,0,719,0,0,0,0,0,0 ,0,727,0,0,0,0,0,733,0,0,0,0,0,739,0,0,0,743,0,0,0, 0,0,0,0,751,0,0,0,0,0,757,0,0,0,761,0,0,0,0,0, 0,0,769,0,0,0,773,0,0,0,0,0,0,0,0,0,0,0,0,0,787,0,0, 0,0,0,0,0,0,0,797,0,0,0,0,0,0,0,0,0,0,0,809,0, 811,0,0,0,0,0,0,0,0,0,821,0,823,0,0,0,827,0,829,0, 0,0,0,0,0,0,0,0,839,0,0,0,0,0,0,0,0,0,0,0,0,0, 853,0,0,0,857,0,859,0,0,0,863,0,0,0,0,0,0,0,0,0,0, 0,0,0,877,0,0,0,881,0,883,0,0,0,887,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,907,0,0,0,911,0,0,0,0,0, 0,0,919,0,0,0,0,0,0,0,0,0,929,0,0,0,0,0,0,0,937, 0,0,0,941,0,0,0,0,0,947,0,0,0,0,0,953,0,0,0,0,0,0,0, 0,0,0,0,0,0,967,0,0,0,971,0,0,0,0,0,977,0,0,0,0, 0,983,0,0,0,0,0,0,0,991,0,0,0,0,0,997,0,0,0,0,0,0,0, 0,0,0,0,1009,0,0,0,1013,0,0,0,0,0,1019,0,1021, 0,0,0,0,0,0,0,0,0,1031,0,1033,0,0,0,0,0,1039,0,0,0, 0,0,0,0,0,0,1049,0,1051,0,0,0,0,0,0,0,0,0,1061, 0,1063,0,0,0,0,0,1069,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,1087,0,0,0,1091,0,1093,0,0,0,1097,0,0,0,0, 0,1103,0,0,0,0,0,1109,0,0,0,0,0,0,0,1117,0,0,0,0,0,1123, 0,0,0,0,0,1129,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,1151,0,1153,0,0,0,0,0,0,0,0,0,1163,0,0, 0,0,0,0,0,1171,0,0,0,0,0,0,0,0,0,1181,0,0,0,0,0, 1187,0,0,0,0,0,1193,0,0,0,0,0,0,0,1201,0,0,0,0,0,0, 0,0,0,0,0,1213,0,0,0,1217,0,0,0,0,0,1223,0,0,0,0, 0,1229,0,1231,0,0,0,0,0,1237,0,0,0,0,0,0,0,0,0,0,0, 1249,0,0,0,0,0,0,0,0,0,1259,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,1277,0,1279,0,0,0,1283,0,0,0,0,0,1289, 0,1291,0,0,0,0,0,1297,0,0,0,1301,0,1303,0,0,0, 1307,0,0,0,0,0,0,0,0,0,0,0,1319,0,1321,0,0,0,0,0,1327, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,1361,0,0,0,0,0,1367,0,0,0,0,0, 1373,0,0,0,0,0,0,0,1381

Borramos Ceros:

DeleteCases [nueva+nueva2+nueva3, 0 ]

{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97, 101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181, 191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277, 281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383, 389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487, 491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601, 607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709, 719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827, 829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947, 953,967,971,977,983,991,997,1009,1013,1019,1021,1031,1033,1039,1049, 1051,1061,1063,1069,1087,1091,1093,1097,1103,1109,1117,1123,1129,1151, 1153,1163,1171,1181,1187,1193,1201,1213,1217,1223,1229,1231,1237,1249, 1259,1277,1279,1283,1289,1291,1297,1301,1303,1307,1319,1321,1327,1361, 1367,1373,1381}

OBTENEMOS LOS PRIMEROS 221 NÚMEROS PRIMOS

AUTOR:

M.M e ING. José de Jesús Camacho Medina

Fresnillo Zacatecas, México - Febrero de 2013