Aritmética Sin Llevar

ARITMÉTICA SIN LLEVAR

Luris Jaén

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II [OTRAS ARITMÉTICAS]

30

[ARITMÉTICA SIN LLEVAR] II

Agradecimientos
 A Dios por darme la oportunidad estudiar, así como de darme la fuerza y la
dedicación que contribuyeron a la culminar este trabajo.
 A mi Esposo José Rosales y a mi Hijo Joseph Rosales por ser mis soportes y
fuentes de inspiración.
 A mis Padres: Luris Lorenzo y Ariel Jaén, que me apoyaron en todo momento.
 Al profesor Jaime Gutiérrez por su asesoramiento científico y estímulo para
seguir creciendo intelectualmente.
 A todos los Profesores de la Licenciatura de Matemáticas.

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DEDICATORIA

A ti mi Querido Dios pues me dirigiste por el mejor camino de mi vida, y me distes la
salud y sabiduría para alcanzar todas mis metas.

A ti esposo querido José, por todo tu amor, compresión y estar siempre mi lado cuando
más lo necesité.

A mi querido hijo Joseph, que es mi fortaleza y mi fuente de inspiración para seguir
adelante.

A mis padres quienes siempre creyeron en mí y me dieron todo el apoyo que necesitaba.

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INTRODUCCIÓN

La Aritmética siempre ha jugado un papel importante en el desarrollo de nuestras
sociedades y en el desarrollo de toda la tecnología que tenemos actualmente.

En el presente trabajo nos centramos en otra nueva Aritmética llamada aritmética sin
llevar en la cual en sus cálculos, sumas y multiplicaciones se llevan a cabo mod 10.
Luego de definir las operaciones elementales en la aritmética sin llevar estudiaremos los
pares sin llevar, primos sin llevar entre otros puntos más.

El objetivo principal de este trabajo, es de realizar un estudio de las propiedades de la
aritmética Sin llevar y de agilizar sus cálculos mediante el uso de la herramienta de
programación Matemática 8.

Además de establecer nuevas definiciones en ésta nueva aritmética haremos también una
comparación entre ésta nueva aritmética y la aritmética clásica.

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Operaciones en la Aritmética Sin Llevar

Suma Sin Llevar

Suma de un Dígito: Dados dos números N y M de un dígito, la suma se realiza como la
suma habitual sólo que aplicando módulo 10.

Ejemplos:

1) 9+ 4 = 3

2) 2) 5 +5 = 0

Suma de n Dígitos: Dados dos números N y M de varios dígitos, la suma se realiza como
la suma habitual sólo que aplicando módulo 10 a cada columna de los sumandos.

Ejemplo:

1) 789 + 376 = 55

Producto Sin Llevar

Producto de un Dígito: Dados dos números N y M de un sólo dígito, el producto se
realiza como el producto habitual sólo que aplicando módulo 10.

Ejemplo:

1) 8 x 9 = 2

Producto de n Dígitos: Dados dos números N y M de varios dígitos, se multiplican sus
factores como en el producto habitual sólo que aplicándoles módulo 10 y luego se realiza
la suma Sin Llevar a los sumandos.

1) 169 x 248=26042

Además dotado de estas operaciones el conjunto de los números enteros es un anillo
unitario y conmutativo.

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Diferencia en la Aritmética Sin Llevar

El negativo de un número en la Aritmética sin llevar es su "10 de complemento ",
obtenido mediante la sustitución de cada dígito diferente de cero por su opuesto módulo
10. Por ejemplo:

1) 650 - 702 = 650 + 308 = 958

2) 2 913 - 21 546 = 2 913 + 89 564 = 81 477

Programación de las Operaciones Sin Llevar

Suma Sin Llevar:

Éste programa calcula la suma de dos números en la Aritmética Sin Llevar.

Se introducen dos números n y m, luego en mayorlong, guardamos la mayor cantidad de
cifras que tiene el número de mayor longitud y luego generamos dos lista a y b, sumamos
aplicando la suma sin llevar entre cada componente de la lista y esa suma la guardamos
en c en forma de lista y luego recuperamos el número con el comando From Digits.

Ejemplo: sumemos 169 con 248

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Producto Sin Llevar:

Éste programa nos calcula el producto de dos números en la Aritmética Sin Llevar.

Introducimos dos números m y n, luego en r y s vamos a guardar la cantidad de cifras de
cada número y en a y b guardaremos los números m y n pero transformados en listas.
Luego haremos el producto Sin Llevar que es parecido al de la Aritmética clásica sólo
que ahora aplicando módulo 10 eso lo guardamos en c y luego en d hacemos ubicamos
los sumandos para luego sumarlos Sin llevar. Hay que tener en cuenta que el resultado de
multiplicar es de tamaño r+s-1 cifras.

Diferencia Sin Llevar:

Éste programa calcula la diferencia de dos números en la Aritmética Sin Llevar.

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Ejemplo: realizamos la resta de 650 con 702.

Opuesto de un Número en la Aritmética Sin Llevar:

Éste programa calcula el opuesto de un número en la Aritmética Sin Llevar.

Ejemplo: Calcular el opuesto de 21 546 en la Aritmética Sin Llevar

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Potencia en la Aritmética Sin Llevar:

Éste programa calcula la potencia en la Aritmética Sin Llevar.

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Pares en la Aritmética Sin Llevar

Definición: Un número entero m es un número par Sin Llevar si y solo si existe otro
número entero n tal que m=21*n

Los pares Sin Llevar son todos los múltiplos de 21. Esto es debido a que 2 es el análogo a
21, el cual es el primo más pequeño en la Aritmética Sin Llevar por ende los múltiplos
de 21 son una opción para definir los números pares en la Aritmética Sin Llevar.

Ejemplos: Los primeros 10 pares en la Aritmética Sin Llevarson los siguientes:
21,42,63,84,5,26,47,68,89,210,…

Pares en la Aritmética Sin Llevar: Nos da una lista de los pares en la Aritmética Sin Llevar de 2,
3 y 4 cifras.

{21,42,63,84,5,26,47,68,89,210,231,252,273,294,215,236,257,278,299,420,441,462,483,404,425,
446,467,488,409,630,651,672,693,614,635,656,677,698,619,840,861,882,803,824,845,866,887,8
08,829,50,71,92,13,34,55,76,97,18,39,260,281,202,223,244,265,286,207,228,249,470,491,412,43
3,454,475,496,417,438,459,680,601,622,643,664,685,606,627,648,669,890,811,832,853,874,895,
816,837,858,879,2100,2121,2142,2163,2184,2105,2126,2147,2168,2189,2310,2331,2352,2373,2
394,2315,2336,2357,2378,2399,2520,2541,2562,2583,2504,2525,2546,2567,2588,2509,2730,275
1,2772,2793,2714,2735,2756,2777,2798,2719,2940,2961,2982,2903,2924,2945,2966,2987,2908,
2929,2150,2171,2192,2113,2134,2155,2176,2197,2118,2139,2360,2381,2302,2323,2344,2365,23
86,2307,2328,2349,2570,2591,2512,2533,2554,2575,2596,2517,2538,2559,2780,2701,2722,2743
,2764,2785,2706,2727,2748,2769,2990,2911,2932,2953,2974,2995,2916,2937,2958,2979,4200,4
221,4242,4263,4284,4205,4226,4247,4268,4289,4410,4431,4452,4473,4494,4415,4436,4457,447
8,4499,4620,4641,4662,4683,4604,4625,4646,4667,4688,4609,4830,4851,4872,4893,4814,4835,
4856,4877,4898,4819,4040,4061,4082,4003,4024,4045,4066,4087,4008,4029,4250,4271,4292,42
13,4234,4255,4276,4297,4218,4239,4460,4481,4402,4423,4444,4465,4486,4407,4428,4449,4670
,4691,4612,4633,4654,4675,4696,4617,4638,4659,4880,4801,4822,4843,4864,4885,4806,4827,4
848,4869,4090,4011,4032,4053,4074,4095,4016,4037,4058,4079,6300,6321,6342,6363,6384,630
5,6326,6347,6368,6389,6510,6531,6552,6573,6594,6515,6536,6557,6578,6599,6720,6741,6762,
6783,6704,6725,6746,6767,6788,6709,6930,6951,6972,6993,6914,6935,6956,6977,6998,6919,61
40,6161,6182,6103,6124,6145,6166,6187,6108,6129,6350,6371,6392,6313,6334,6355,6376,6397
,6318,6339,6560,6581,6502,6523,6544,6565,6586,6507,6528,6549,6770,6791,6712,6733,6754,6
775,6796,6717,6738,6759,6980,6901,6922,6943,6964,6985,6906,6927,6948,6969,6190,6111,613
2,6153,6174,6195,6116,6137,6158,6179,8400,8421,8442,8463,8484,8405,8426,8447,8468,8489,
8610,8631,8652,8673,8694,8615,8636,8657,8678,8699,8820,8841,8862,8883,8804,8825,8846,88
67,8888,8809,8030,8051,8072,8093,8014,8035,8056,8077,8098,8019,8240,8261,8282,8203,8224
,8245,8266,8287,8208,8229,8450,8471,8492,8413,8434,8455,8476,8497,8418,8439,8660,8681,8
602,8623,8644,8665,8686,8607,8628,8649,8870,8891,8812,8833,8854,8875,8896,8817,8838,885

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9,8080,8001,8022,8043,8064,8085,8006,8027,8048,8069,8290,8211,8232,8253,8274,8295,8216,
8237,8258,8279,500,521,542,563,584,505,526,547,568,589,710,731,752,773,794,715,736,757,77
8,799,920,941,962,983,904,925,946,967,988,909,130,151,172,193,114,135,156,177,198,119,340,
361,382,303,324,345,366,387,308,329,550,571,592,513,534,555,576,597,518,539,760,781,702,7
23,744,765,786,707,728,749,970,991,912,933,954,975,996,917,938,959,180,101,122,143,164,18
5,106,127,148,169,390,311,332,353,374,395,316,337,358,379,2600,2621,2642,2663,2684,2605,2
626,2647,2668,2689,2810,2831,2852,2873,2894,2815,2836,2857,2878,2899,2020,2041,2062,208
3,2004,2025,2046,2067,2088,2009,2230,2251,2272,2293,2214,2235,2256,2277,2298,2219,2440,
2461,2482,2403,2424,2445,2466,2487,2408,2429,2650,2671,2692,2613,2634,2655,2676,2697,26
18,2639,2860,2881,2802,2823,2844,2865,2886,2807,2828,2849,2070,2091,2012,2033,2054,2075
,2096,2017,2038,2059,2280,2201,2222,2243,2264,2285,2206,2227,2248,2269,2490,2411,2432,2
453,2474,2495,2416,2437,2458,2479,4700,4721,4742,4763,4784,4705,4726,4747,4768,4789,491
0,4931,4952,4973,4994,4915,4936,4957,4978,4999,4120,4141,4162,4183,4104,4125,4146,4167,
4188,4109,4330,4351,4372,4393,4314,4335,4356,4377,4398,4319,4540,4561,4582,4503,4524,45
45,4566,4587,4508,4529,4750,4771,4792,4713,4734,4755,4776,4797,4718,4739,4960,4981,4902
,4923,4944,4965,4986,4907,4928,4949,4170,4191,4112,4133,4154,4175,4196,4117,4138,4159,4
380,4301,4322,4343,4364,4385,4306,4327,4348,4369,4590,4511,4532,4553,4574,4595,4516,453
7,4558,4579,6800,6821,6842,6863,6884,6805,6826,6847,6868,6889,6010,6031,6052,6073,6094,
6015,6036,6057,6078,6099,6220,6241,6262,6283,6204,6225,6246,6267,6288,6209,6430,6451,64
72,6493,6414,6435,6456,6477,6498,6419,6640,6661,6682,6603,6624,6645,6666,6687,6608,6629
,6850,6871,6892,6813,6834,6855,6876,6897,6818,6839,6060,6081,6002,6023,6044,6065,6086,6
007,6028,6049,6270,6291,6212,6233,6254,6275,6296,6217,6238,6259,6480,6401,6422,6443,646
4,6485,6406,6427,6448,6469,6690,6611,6632,6653,6674,6695,6616,6637,6658,6679,8900,8921,
8942,8963,8984,8905,8926,8947,8968,8989,8110,8131,8152,8173,8194,8115,8136,8157,8178,81
99,8320,8341,8362,8383,8304,8325,8346,8367,8388,8309,8530,8551,8572,8593,8514,8535,8556
,8577,8598,8519,8740,8761,8782,8703,8724,8745,8766,8787,8708,8729,8950,8971,8992,8913,8
934,8955,8976,8997,8918,8939,8160,8181,8102,8123,8144,8165,8186,8107,8128,8149,8370,839
1,8312,8333,8354,8375,8396,8317,8338,8359,8580,8501,8522,8543,8564,8585,8506,8527,8548,
8569,8790,8711,8732,8753,8774,8795,8716,8737,8758,8779}

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Cuadrados en la Aritmética Sin Llevar

Los cuadrados Carryless son de la forma n x n. Para n = 0, 1, 2, 3 obtenemos 0, 1, 4, 9,
para n> 3 tenemos los siguientes cuadrados en la aritmética sin llevar: 4 x 4 = 6, 5 x 5 =
5, 6 x 6 = 6, 7 x 7 = 9, 8 x 8 = 4, 9 x 9 = 1, 10 x 10 = 100,. . ., Dando la secuencia :
0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 100, 121, 144, 169, 186, 105, 126, 149, 164,. .
Estos cuadrados fueron aportados por Henry Bottomley en 20 de febrero 2001,

Programación Utilizando Matemática 8.0

La siguiente función nos da una lista de los 100 primeros cuadrados en la Aritmética Sin
Llevar.

{1,4,9,6,5,6,9,4,1,100,121,144,169,186,105,126,149,164,181,400,441,484,429,466,405,4
46,489,424,461,900,961,924,989,946,905,966,929,984,941,600,681,664,649,626,605,686
,669,644,621,500,501,504,509,506,505,506,509,504,501,600,621,644,669,686,605,626,6
49,664,681,900,941,984,929,966,905,946,989,924,961,400,461,424,489,446,405,466,429
,484,441,100,181,164,149,126,105,186,169,144,121,10000}

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Problemas Algebrizados

El secreto para entender la aritmética Sin Llevar es introducir un poco de álgebra. Vamos
a denotar Z10 el anillo de enteros mod 10 y Z10 [X] el anillo de polinomios en X con
coeficientes en Z10. Entonces se puede representar números en la Aritmética Sin Llevar
por elementos del anillo de polinomios Z10[X]: por ejemplo 21 corresponde a 2X+1,109
corresponde a X2+9, y así sucesivamente. Si tenemos la siguiente suma: 785+376=51.La
misma corresponde a (7x2+8x+5)+ (3X2+7X+6)=5x+1, donde los polinomios se suman
o multiplican de la manera habitual, y luego a los coeficientes se les aplica módulo 10.

En el anillo Z10[X] no sólo se puede sumar y multiplicar, también podemos restar, Los
negativos de los elementos de Z10 son-1=9,-2=8.. . ,-9=1,y lo mismo para los elementos
de Z10[X].Como vimos el negativo de un número es su complemento módulo 10.

Función F10: La función F10 "transforma el número n en un polinomio de Z10[X].

Ejemplo: Transformar 785 en un polinomio de Z10[X].

Función Mod 10: La función Mod10 "nos da los coeficientes del polinomio de Z10[x].

Ejemplo: Obtener los coeficientes del polinomio de Z10[x].

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Operaciones Utilizando Algebra.

Utilizando el álgebra podemos realizar las mismas operaciones elementale presentadas
anteriormente.

Cuadrados en la Aritmética Sin Llevar

{1,4,9,6,5,6,9,4,1,100,121,144,169,186,105,126,149,164,181,400,441,484,429,466,405,4
46,489,424,461,900,961,924,989,946,905,966,929,984,941,600,681,664,649,626,605,686
,669,644,621,500,501,504,509,506,505,506,509,504,501,600,621,644,669,686,605,626,6
49,664,681,900,941,984,929,966,905,946,989,924,961,400,461,424,489,446,405,466,429
,484,441,100,181,164,149,126,105,186,169,144,121,10000}

Opuesto de un Número en la Aritmética Sin Llevar:

Éste programa nos calcula el opuesto de un número.

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Pares en la Aritmética Sin Llevar:

{21,42,63,84,5,26,47,68,89,210,231,252,273,294,215,236,257,278,299,420,441,462,483,
404,425,446,467,488,409,630,651,672,693,614,635,656,677,698,619,840,861,882,803,82
4,845,866,887,808,829,50,71,92,13,34,55,76,97,18,39,260,281,202,223,244,265,286,207,
228,249,470,491,412,433,454,475,496,417,438,459,680,601,622,643,664,685,606,627,64
8,669,890,811,832,853,874,895,816,837,858,879,2100,2121,2142,2163,2184,2105,2126,
2147,2168,2189,2310,2331,2352,2373,2394,2315,2336,2357,2378,2399,2520,2541,2562,
2583,2504,2525,2546,2567,2588,2509,2730,2751,2772,2793,2714,2735,2756,2777,2798,
2719,2940,2961,2982,2903,2924,2945,2966,2987,2908,2929,2150,2171,2192,2113,2134,
2155,2176,2197,2118,2139,2360,2381,2302,2323,2344,2365,2386,2307,2328,2349,2570,
2591,2512,2533,2554,2575,2596,2517,2538,2559,2780,2701,2722,2743,2764,2785,2706,
2727,2748,2769,2990,2911,2932,2953,2974,2995,2916,2937,2958,2979,4200,4221,4242,
4263,4284,4205,4226,4247,4268,4289,4410,4431,4452,4473,4494,4415,4436,4457,4478,
4499,4620,4641,4662,4683,4604,4625,4646,4667,4688,4609,4830,4851,4872,4893,4814,
4835,4856,4877,4898,4819,4040,4061,4082,4003,4024,4045,4066,4087,4008,4029,4250,
4271,4292,4213,4234,4255,4276,4297,4218,4239,4460,4481,4402,4423,4444,4465,4486,
4407,4428,4449,4670,4691,4612,4633,4654,4675,4696,4617,4638,4659,4880,4801,4822,
4843,4864,4885,4806,4827,4848,4869,4090,4011,4032,4053,4074,4095,4016,4037,4058,
4079,6300,6321,6342,6363,6384,6305,6326,6347,6368,6389,6510,6531,6552,6573,6594,
6515,6536,6557,6578,6599,6720,6741,6762,6783,6704,6725,6746,6767,6788,6709,6930,

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6951,6972,6993,6914,6935,6956,6977,6998,6919,6140,6161,6182,6103,6124,6145,6166,
6187,6108,6129,6350,6371,6392,6313,6334,6355,6376,6397,6318,6339,6560,6581,6502,
6523,6544,6565,6586,6507,6528,6549,6770,6791,6712,6733,6754,6775,6796,6717,6738,
6759,6980,6901,6922,6943,6964,6985,6906,6927,6948,6969,6190,6111,6132,6153,6174,
6195,6116,6137,6158,6179,8400,8421,8442,8463,8484,8405,8426,8447,8468,8489,8610,
8631,8652,8673,8694,8615,8636,8657,8678,8699,8820,8841,8862,8883,8804,8825,8846,
8867,8888,8809,8030,8051,8072,8093,8014,8035,8056,8077,8098,8019,8240,8261,8282,
8203,8224,8245,8266,8287,8208,8229,8450,8471,8492,8413,8434,8455,8476,8497,8418,
8439,8660,8681,8602,8623,8644,8665,8686,8607,8628,8649,8870,8891,8812,8833,8854,
8875,8896,8817,8838,8859,8080,8001,8022,8043,8064,8085,8006,8027,8048,8069,8290,
8211,8232,8253,8274,8295,8216,8237,8258,8279,500,521,542,563,584,505,526,547,568,
589,710,731,752,773,794,715,736,757,778,799,920,941,962,983,904,925,946,967,988,90
9,130,151,172,193,114,135,156,177,198,119,340,361,382,303,324,345,366,387,308,329,
550,571,592,513,534,555,576,597,518,539,760,781,702,723,744,765,786,707,728,749,97
0,991,912,933,954,975,996,917,938,959,180,101,122,143,164,185,106,127,148,169,390,
311,332,353,374,395,316,337,358,379,2600,2621,2642,2663,2684,2605,2626,2647,2668,
2689,2810,2831,2852,2873,2894,2815,2836,2857,2878,2899,2020,2041,2062,2083,2004,
2025,2046,2067,2088,2009,2230,2251,2272,2293,2214,2235,2256,2277,2298,2219,2440,
2461,2482,2403,2424,2445,2466,2487,2408,2429,2650,2671,2692,2613,2634,2655,2676,
2697,2618,2639,2860,2881,2802,2823,2844,2865,2886,2807,2828,2849,2070,2091,2012,
2033,2054,2075,2096,2017,2038,2059,2280,2201,2222,2243,2264,2285,2206,2227,2248,
2269,2490,2411,2432,2453,2474,2495,2416,2437,2458,2479,4700,4721,4742,4763,4784,
4705,4726,4747,4768,4789,4910,4931,4952,4973,4994,4915,4936,4957,4978,4999,4120,
4141,4162,4183,4104,4125,4146,4167,4188,4109,4330,4351,4372,4393,4314,4335,4356,
4377,4398,4319,4540,4561,4582,4503,4524,4545,4566,4587,4508,4529,4750,4771,4792,
4713,4734,4755,4776,4797,4718,4739,4960,4981,4902,4923,4944,4965,4986,4907,4928,
4949,4170,4191,4112,4133,4154,4175,4196,4117,4138,4159,4380,4301,4322,4343,4364,
4385,4306,4327,4348,4369,4590,4511,4532,4553,4574,4595,4516,4537,4558,4579,6800,
6821,6842,6863,6884,6805,6826,6847,6868,6889,6010,6031,6052,6073,6094,6015,6036,
6057,6078,6099,6220,6241,6262,6283,6204,6225,6246,6267,6288,6209,6430,6451,6472,
6493,6414,6435,6456,6477,6498,6419,6640,6661,6682,6603,6624,6645,6666,6687,6608,
6629,6850,6871,6892,6813,6834,6855,6876,6897,6818,6839,6060,6081,6002,6023,6044,
6065,6086,6007,6028,6049,6270,6291,6212,6233,6254,6275,6296,6217,6238,6259,6480,
6401,6422,6443,6464,6485,6406,6427,6448,6469,6690,6611,6632,6653,6674,6695,6616,
6637,6658,6679,8900,8921,8942,8963,8984,8905,8926,8947,8968,8989,8110,8131,8152,
8173,8194,8115,8136,8157,8178,8199,8320,8341,8362,8383,8304,8325,8346,8367,8388,
8309,8530,8551,8572,8593,8514,8535,8556,8577,8598,8519,8740,8761,8782,8703,8724,
8745,8766,8787,8708,8729,8950,8971,8992,8913,8934,8955,8976,8997,8918,8939,8160,
8181,8102,8123,8144,8165,8186,8107,8128,8149,8370,8391,8312,8333,8354,8375,8396,

45


8317,8338,8359,8580,8501,8522,8543,8564,8585,8506,8527,8548,8569,8790,8711,8732,
8753,8774,8795,8716,8737,8758,8779}

Primos en la Aritmética Sin Llevar

Primos sin Llevar

En ésta aritmética además del 1 existen otros números capaces de dividirlo, por ejemplo:
Desde el 1 x 1 = 1, 3 x7 = 1 9 x 9 = 1, todos éstos números 1, 3, 7 y 9 dividen a 1 y así
también a cualquier número.

Si queremos dar una definición de primos en la aritmética sin llevar debemos saber que
éste tipo de primos no tienen una única forma de factorización.

Podemos definir un primo Carryless de la siguiente manera: “Un primo Carryless es un
número cuyas factorizaciones sólo son de la forma π = u x p donde u ϵ {1,3,7,9}

21=21 x 1 21=47 x 3 21= 63 x 7 21= 89 x 9

23= 23 x 1 23= 41 x 3 23= 69 x 7 23= 87x 9

25= 25 x 1 25= 45 x 3 25= 65 x 7 25= 85 x 9

27= 27 x 1 27= 49 x 3 27= 61 x 7 27= 83 x 9

¿Cuáles son los elementos irreductibles f10 (x) ϵ Z10 [X]? Si f10 (X) ↔ [f2 (X), f5 (X)]
es irreducible entonces, f2 y f5 uno tendrá que ser la unidad y el otro el elemento
irreducible, porque si f2 = g2h2 entonces tenemos la factorización [f2, f5] = [g2, f5] [h2,
1]. También [f2, f5] = [f2, 1] [1, f5], así que uno de los f2, f5 debe ser irreductible y el
otro debe ser una unidad. Así que los elementos irreductibles en Z10 [X] son de la forma
[f2 (X), u], donde f2 (X) es un polinomio irreducible mod 2 de grado≥1 y u ϵ {1, 2, 3,

46


4}, junto con elementos de la forma [1, f5 (X)], donde f5 (X) es un polinomio irreducible
modulo 5 de grado ≥ 1.

Los polinomios irreducibles mod 2 son X, X + 1, X2 + X +1,. . ., Y los polinomios
irreducibles mod 5 son uX, uX + v,. . ., Donde u, v ϵ{1, 2, 3, 4}

Los elementos irreducibles [1, f5 (X)] son aquellos primos llamados tipo-e

21, 23, 25, 27, 29, 41, 43, 45, 47, 49, 61, 63, 65, 67, 69, 81, 83, 85, 87, 89, 201, 209, 227,
229, 241, 243, 261, 263, 287, 289, 403, 407,…

Los elementos irreducibles [f2 (X), u] son aquellos primos llamados tipo-f

51, 52, 53, 54, 56, 57, 58, 59, 551, 553, 557, 559, 5051, 5053, 5057, 5059, 5501, 5503,
5507, 5509, 50051, 50053, 50057, 50059, 55001,…

PRIMOS SIN LLEVAR = PRIMO TIPO-E U PRIMO TIPO-F

El siguiente programa nos permite saber si dado un número n es primo o no en la
Aritmética Sin Llevar.

Primos Sin Llevar hasta seis cifras.

PrimosSinLlevar={21,23,25,27,29,41,43,45,47,49,51,52,53,54,56,57,58,59,61,63,65,67,6
9,81,83,85,87,89,201,209,227,229,241,243,261,263,287,289,403,407,421,427,443,449,46
3,469,481,487,551,553,557,559,603,607,623,629,641,647,661,667,683,689,801,809,821,
823,847,849,867,869,881,883,2023,2027,2043,2047,2061,2069,2081,2089,2207,2209,22
21,2223,2263,2267,2281,2287,2401,2407,2421,2423,2441,2449,2483,2489,2603,2609,26
27,2629,2641,2649,2681,2687,2801,2803,2827,2829,2863,2867,2883,2889,4023,4027,40

47


41,4049,4063,4067,4081,4089,4201,4203,4243,4249,4267,4269,4283,4287,4403,4409,44
21,4429,4441,4447,4467,4469,4601,4607,4621,4629,4643,4649,4661,4663,4807,4809,48
41,4847,4861,4863,4883,4887,5051,5053,5057,5059,5501,5503,5507,5509,6021,6029,60
43,6047,6061,6069,6083,6087,6201,6203,6223,6227,6247,6249,6263,6269,6403,6409,64
47,6449,6461,6467,6481,6489,6601,6607,6641,6643,6663,6669,6681,6689,6807,6809,68
23,6827,6841,6843,6861,6867,8021,8029,8041,8049,8063,8067,8083,8087,8207,8209,82
21,8227,8243,8247,8281,8283,8401,8407,8423,8429,8461,8469,8481,8483,8603,8609,86
21,8627,8661,8669,8687,8689,8801,8803,8823,8829,8843,8847,8887,8889,20001,20009,
20023,20043,20063,20083,20209,20227,20241,20249,20261,20269,20287,20401,20441,
20447,20461,20467,20601,20621,20627,20681,20687,20809,20821,20829,20847,20867,
20881,20889,22003,22021,22041,22043,22069,22087,22089,22207,22221,22223,22243,
22261,22289,22409,22421,22429,22447,22449,22463,22483,22601,22607,22647,22683,
22689,22809,22823,22827,22881,22887,24003,24021,24023,24047,24049,24061,24089,
24209,24241,24247,24263,24267,24401,24407,24427,24443,24449,24609,24627,24629,
24643,24661,24669,24683,24807,24823,24849,24861,24863,24881,26003,26029,26041,
26067,26069,26081,26083,26209,26243,26247,26261,26267,26401,26407,26463,26469,
26487,26609,26623,26641,26649,26663,26687,26689,26807,26821,26841,26843,26869,
26883,28003,28027,28029,28049,28061,28063,28081,28207,28229,28241,28263,28281,
28283,28409,28423,28443,28467,28469,28481,28489,28601,28607,28623,28629,28667,
28809,28821,28827,28883,28887,40003,40007,40021,40041,40061,40081,40207,40247,
40249,40267,40269,40403,40423,40427,40449,40469,40483,40487,40603,40629,40643,
40647,40663,40667,40689,40807,40827,40829,40887,40889,42001,42023,42029,42043,
42061,42067,42087,42203,42221,42241,42263,42269,42283,42287,42403,42427,42429,
42481,42489,42609,42623,42647,42661,42681,42687,42807,42809,42821,42823,42869,
44001,44023,44047,44063,44069,44081,44087,44207,44209,44261,44263,44289,44409,
44427,44441,44447,44463,44481,44603,44641,44649,44667,44669,44803,44821,44843,
44847,44861,44883,44889,46001,46021,46027,46043,46049,46067,46083,46207,46209,
46229,46241,46243,46409,46421,46443,46461,46467,46487,46603,46647,46649,46661,
46669,46803,46823,46829,46841,46863,46867,46881,48001,48027,48041,48047,48063,
48083,48089,48203,48223,48227,48243,48249,48261,48281,48403,48421,48429,48487,
48489,48609,48621,48627,48641,48667,48683,48807,48809,48849,48881,48883,50051,
50053,50057,50059,55001,55003,55007,55009,55551,55553,55557,55559,60003,60007,
60029,60049,60069,60089,60203,60221,60223,60281,60283,60407,60421,60443,60447,
60463,60467,60481,60607,60623,60627,60641,60661,60683,60687,60803,60841,60843,
60861,60863,62009,62021,62027,62047,62063,62069,62083,62201,62203,62227,62229,
62261,62401,62427,62443,62469,62483,62489,62607,62621,62623,62681,62689,62807,
62829,62849,62861,62867,62883,62887,64009,64027,64043,64061,64067,64083,64089,
64207,64229,64243,64247,64269,64281,64287,64407,64441,64449,64461,64463,64601,
64623,64643,64649,64667,64689,64801,64803,64867,64869,64881,66009,66023,66029,
66041,66047,66063,66087,66207,66221,66227,66249,66263,66267,66289,66407,66441,

48


66443,66461,66469,66601,66629,66647,66663,66669,66683,66801,66803,66821,66847,
66849,68009,68023,68043,68049,68067,68081,68087,68201,68203,68241,68287,68289,
68401,68423,68429,68449,68463,68487,68607,68621,68629,68681,68683,68807,68823,
68827,68841,68847,68869,68889,80001,80009,80027,80047,80067,80087,80201,80221,
80229,80243,80263,80281,80289,80409,80423,80429,80483,80489,80609,80643,80649,
80663,80669,80801,80823,80841,80849,80861,80869,80883,82007,82029,82047,82049,
82061,82081,82083,82201,82223,82227,82283,82289,82403,82409,82443,82481,82487,
82601,82621,82629,82641,82643,82667,82687,82803,82827,82829,82847,82869,82881,
84007,84027,84029,84041,84043,84069,84081,84203,84227,84241,84267,84269,84289,
84401,84421,84423,84447,84461,84469,84487,84603,84609,84623,84641,84647,84801,
84843,84849,84863,84867,86007,86021,86049,86061,86063,86087,86089,86203,86229,
86247,86249,86261,86287,86401,86427,86441,86449,86467,86481,86483,86603,86609,
86661,86667,86683,86801,86843,86847,86863,86869,88007,88021,88023,88041,88067,
88069,88089,88201,88223,88229,88283,88287,88403,88409,88421,88427,88463,88601,
88627,88647,88661,88663,88681,88689,88803,88821,88849,88867,88887,88889,200081
,200083,200087,200089,200209,200223,200243,200269,200281,200407,200429,200449,
200467,200483,200603,200621,200641,200663,200687,200801,200827,200847,200861,
200889,202041,202049,202061,202069,202201,202203,202287,202289,202401,202403,
202421,202429,202607,202609,202621,202629,202807,202809,202881,202883,204003,
204007,204021,204029,204043,204047,204223,204229,204247,204249,204263,204267,
204401,204409,204421,204427,204483,204489,204601,204609,204623,204629,204681,
204687,204821,204827,204841,204843,204863,204867,206001,206009,206023,206027,
206041,206049,206203,206207,206227,206229,206281,206283,206427,206429,206441,
206447,206461,206469,206621,206623,206643,206649,206661,206669,206803,206807,
206821,206823,206887,206889,208043,208047,208063,208067,208201,208207,208223,
208227,208403,208409,208481,208487,208601,208607,208683,208689,208803,208809,
208823,208827,220003,220023,220087,220241,220267,220283,220289,220409,220421,
220469,220481,220483,220601,220607,220649,220661,220667,220681,220801,220827,
220847,220869,220883,220887,220889,222001,222021,222043,222047,222049,222069,
222083,222209,222223,222241,222429,222443,222463,222467,222607,222627,222629,
222643,222663,222803,222807,222841,222861,222867,222889,224029,224041,224063,
224069,224203,224221,224261,224281,224289,224407,224423,224429,224463,224481,
224487,224607,224621,224623,224627,224641,224663,224689,224849,224861,224887,
226003,226009,226063,226083,226201,226227,226263,226281,226289,226427,226443,
226449,226467,226481,226487,226603,226629,226641,226643,226647,226669,226689,
226829,226849,226863,228007,228023,228027,228029,228043,228061,228083,228209,
228269,228287,228403,228407,228461,228489,228623,228647,228649,228667,228681,
228801,228807,228843,228863,228867,228889,240001,240021,240089,240203,240227,
240263,240281,240287,240407,240429,240449,240463,240481,240483,240489,240647,
240669,240681,240683,240807,240809,240843,240867,240869,240887,242009,242021,

49


242023,242029,242041,242067,242081,242201,242209,242267,242283,242407,242409,
242441,242461,242469,242483,242603,242663,242689,242821,242843,242849,242869,
242887,244001,244003,244061,244081,244229,244241,244243,244269,244287,244289,
244423,244443,244461,244607,244629,244661,244683,244687,244801,244823,244841,
244847,244849,244863,244883,246023,246047,246061,246063,246209,246221,246223,
246261,246287,246289,246443,246467,246489,246601,246627,246667,246683,246687,
246809,246821,246827,246829,246847,246861,246883,248007,248027,248041,248043,
248049,248063,248081,248223,248241,248261,248269,248401,248409,248447,248467,
248469,248483,248603,248621,248647,248809,248823,248829,248841,248861,260009,
260029,260081,260201,260203,260247,260261,260263,260283,260443,260461,260487,
260489,260603,260621,260641,260667,260681,260687,260689,260807,260823,260867,
260883,260889,262001,262021,262027,262029,262049,262063,262089,262229,262241,
262247,262261,262283,262407,262467,262481,262601,262603,262649,262661,262669,
262687,262801,262809,262863,262887,264007,264009,264069,264089,264209,264227,
264241,264243,264249,264267,264287,264403,264421,264469,264483,264487,264627,
264647,264669,264821,264847,264849,264861,264881,264883,266027,266043,266067,
266069,266201,266221,266223,266229,266243,266269,266287,266409,266423,266463,
266483,266487,266647,266663,266681,266801,266827,266829,266869,266881,266883,
268003,268023,268041,268047,268049,268067,268089,268201,268221,268227,268249,
268269,268407,268429,268443,268601,268609,268643,268661,268663,268687,268827,
268849,268861,268869,280007,280027,280083,280209,280223,280243,280261,280281,
280283,280287,280403,280409,280441,280463,280469,280489,280601,280629,280661,
280687,280689,280849,280863,280881,280887,282009,282029,282041,282043,282047,
282061,282087,282203,282207,282249,282263,282269,282281,282403,282421,282423,
282447,282467,282621,282647,282663,282667,282801,282827,282849,284021,284049,
284061,284067,284241,284269,284283,284403,284423,284427,284429,284449,284467,
284481,284603,284621,284627,284667,284683,284689,284807,284829,284869,284881,
284889,286001,286007,286067,286087,286221,286241,286267,286407,286421,286443,
286447,286449,286461,286481,286623,286641,286647,286663,286683,286689,286809,
286823,286867,286881,286889,288003,288021,288023,288027,288047,288069,288087,
288203,288209,288247,288263,288267,288281,288427,288441,288443,288463,288489,
288603,288607,288669,288681,288801,288861,288883,400061,400063,400067,400069,
400201,400221,400247,400269,400287,400403,400423,400441,400467,400481,400607,
400627,400649,400663,400689,400809,400829,400843,400861,400883,402003,402007,
402041,402049,402083,402087,402223,402227,402241,402247,402281,402283,402401,
402409,402443,402449,402461,402467,402601,402609,402641,402647,402663,402669,
402823,402827,402843,402849,402887,402889,404023,404027,404083,404087,404203,
404209,404243,404247,404401,404407,404463,404469,404603,404609,404661,404667,
404801,404807,404843,404847,406021,406029,406081,406089,406207,406209,406261,
406263,406407,406409,406441,406449,406601,406603,406641,406649,406801,406803,

50


406867,406869,408001,408009,408043,408047,408081,408089,408203,408207,408241,
408243,408267,408269,408421,408429,408441,408443,408483,408489,408621,408629,
408647,408649,408681,408687,408803,408807,408847,408849,408861,408863,420003,
420043,420067,420201,420229,420247,420263,420267,420269,420287,420401,420407,
420421,420427,420461,420489,420609,420629,420641,420661,420663,420827,420863,
420869,420881,422023,422029,422049,422081,422221,422267,422289,422407,422423,
422441,422443,422447,422469,422481,422607,422623,422643,422649,422661,422667,
422803,422821,422841,422861,422869,424007,424021,424043,424047,424049,424063,
424083,424201,424207,424223,424227,424269,424283,424427,424443,424461,424487,
424489,424603,424607,424621,424669,424809,424829,424867,426001,426029,426041,
426063,426083,426087,426089,426203,426207,426221,426227,426269,426281,426407,
426423,426447,426449,426483,426623,426627,426649,426683,426809,426843,426881,
428003,428009,428023,428063,428223,428249,428289,428403,428429,428449,428469,
428481,428483,428487,428627,428647,428661,428667,428683,428689,428801,428823,
428847,428861,428869,440001,440041,440069,440207,440209,440227,440229,440267,
440283,440429,440461,440463,440487,440607,440623,440649,440661,440663,440669,
440689,440803,440823,440847,440861,440867,442001,442003,442021,442061,442201,
442223,442243,442263,442281,442287,442289,442407,442421,442449,442463,442467,
442621,442643,442683,442829,442849,442867,442869,442881,442883,444007,444023,
444047,444061,444081,444083,444089,444209,444221,444243,444249,444281,444403,
444441,444487,444601,444609,444627,444629,444663,444687,444821,444829,444843,
444881,446009,446027,446041,446043,446049,446061,446081,446229,446241,446267,
446283,446289,446403,446423,446469,446607,446609,446621,446629,446663,446681,
446801,446809,446827,446863,448021,448023,448043,448087,448209,448221,448241,
448247,448249,448263,448287,448401,448427,448447,448463,448467,448627,448669,
448683,448809,448821,448841,448843,448867,448869,460009,460049,460061,460207,
460227,460243,460263,460269,460403,460427,460441,460461,460467,460469,460481,
460621,460667,460669,460683,460801,460803,460821,460823,460863,460887,462007,
462009,462029,462069,462221,462241,462261,462263,462287,462289,462429,462447,
462487,462603,462629,462641,462663,462667,462809,462827,462847,462867,462881,
462883,462889,464003,464027,464043,464069,464081,464087,464089,464221,464229,
464247,464289,464401,464409,464421,464423,464467,464483,464607,464649,464683,
464801,464829,464841,464847,464889,466001,466023,466041,466047,466049,466069,
466089,466201,466209,466223,466267,466401,466403,466421,466429,466467,466489,
466607,466627,466661,466821,466849,466863,466881,466887,468027,468029,468047,
468083,468201,468229,468247,468249,468261,468263,468423,468461,468487,468609,
468623,468643,468663,468667,468801,468829,468841,468843,468849,468867,468883,
480007,480047,480063,480223,480261,480267,480289,480401,480421,480449,480467,
480469,480603,480609,480623,480629,480669,480681,480809,480821,480843,480861,
480863,480867,480883,482021,482027,482041,482089,482207,482229,482249,482261,

51


482269,482403,482427,482441,482447,482463,482469,482603,482627,482643,482647,
482649,482661,482689,482829,482863,482881,484003,484029,484041,484043,484047,
484067,484087,484201,484221,484263,484403,484407,484429,484461,484623,484647,
484669,484681,484683,484803,484809,484823,484827,484861,484887,486009,486021,
486049,486067,486081,486083,486087,486201,486247,486289,486423,486427,486441,
486487,486603,486627,486641,486643,486687,486803,486807,486823,486829,486861,
486889,488001,488007,488027,488067,488209,488227,488243,488261,488269,488423,
488443,488463,488469,488481,488487,488607,488621,488641,488661,488683,488687,
488689,488827,488841,488881,500501,500503,500507,500509,505001,505003,505007,
505009,505551,505553,505557,505559,550551,550553,550557,550559,555051,555053,
555057,555059,555501,555503,555507,555509,600041,600043,600047,600049,600201,
600227,600249,600267,600281,600403,600421,600447,600461,600483,600607,600629,
600643,600669,600687,600809,600823,600841,600863,600889,602001,602009,602021,
602029,602063,602067,602203,602207,602247,602249,602261,602263,602423,602429,
602461,602463,602481,602489,602621,602627,602667,602669,602681,602689,602803,
602807,602841,602843,602867,602869,604021,604029,604081,604089,604207,604209,
604241,604243,604407,604409,604461,604469,604601,604603,604661,604669,604801,
604803,604847,604849,606023,606027,606083,606087,606203,606209,606263,606267,
606401,606407,606443,606449,606603,606609,606641,606647,606801,606807,606863,
606867,608003,608007,608023,608027,608061,608069,608221,608223,608261,608267,
608283,608287,608401,608409,608441,608447,608463,608469,608601,608609,608643,
608649,608661,608667,608827,608829,608863,608869,608883,608887,620003,620047,
620063,620201,620227,620243,620247,620249,620267,620289,620401,620407,620429,
620441,620481,620487,620609,620641,620643,620661,620689,620821,620843,620849,
620887,622003,622009,622043,622083,622229,622269,622283,622403,622421,622423,
622427,622449,622469,622489,622623,622629,622641,622647,622667,622687,622801,
622841,622849,622867,622883,624001,624023,624027,624029,624043,624061,624089,
624203,624207,624221,624249,624281,624287,624407,624423,624467,624469,624483,
624623,624669,624683,624687,624809,624821,624863,626007,626023,626043,626063,
626067,626069,626081,626201,626207,626223,626249,626283,626287,626427,626429,
626441,626463,626487,626603,626607,626649,626681,626809,626847,626889,628021,
628069,628083,628089,628229,628247,628281,628407,628421,628449,628461,628463,
628467,628483,628607,628641,628647,628663,628669,628683,628803,628841,628849,
628861,628881,640001,640049,640061,640207,640209,640223,640247,640287,640289,
640427,640441,640443,640489,640607,640629,640641,640643,640649,640669,640683,
640803,640841,640847,640867,640883,642027,642063,642081,642083,642209,642227,
642243,642261,642267,642269,642281,642401,642443,642447,642467,642487,642623,
642649,642687,642809,642847,642849,642861,642863,642881,644009,644021,644041,
644061,644063,644069,644087,644223,644229,644247,644261,644289,644403,644449,
644483,644607,644609,644621,644643,644681,644689,644801,644809,644843,644887,

52


646007,646021,646023,646029,646041,646067,646083,646209,646221,646263,646269,
646281,646403,646427,646461,646601,646609,646627,646643,646687,646689,646821,
646863,646881,646889,648001,648003,648041,648081,648201,648221,648227,648229,
648243,648263,648283,648407,648443,648447,648469,648481,648623,648663,648681,
648821,648823,648847,648849,648869,648889,660009,660041,660069,660207,660243,
660249,660263,660287,660403,660421,660441,660447,660449,660461,660487,660623,
660647,660649,660681,660801,660803,660827,660843,660881,660883,662023,662067,
662087,662089,662201,662241,662243,662267,662269,662289,662427,662441,662483,
662609,662643,662647,662663,662683,662801,662823,662847,662861,662863,662869,
662889,664001,664029,664049,664061,664067,664069,664083,664201,664209,664247,
664283,664401,664403,664429,664447,664481,664489,664607,664641,664687,664821,
664827,664843,664869,664881,666003,666021,666027,666029,666049,666063,666087,
666229,666267,666281,666289,666401,666409,666423,666447,666481,666483,666607,
666623,666669,666801,666829,666861,666867,666889,668007,668009,668049,668089,
668227,668229,668241,668243,668261,668281,668427,668467,668489,668603,668643,
668647,668661,668689,668809,668821,668823,668829,668847,668867,668887,680007,
680043,680067,680229,680241,680247,680283,680401,680447,680449,680469,680481,
680603,680609,680621,680649,680683,680689,680809,680823,680841,680843,680847,
680863,680881,682001,682007,682047,682087,682209,682241,682249,682263,682287,
682421,682427,682443,682449,682463,682483,682607,682623,682627,682629,682641,
682661,682681,682821,682861,682887,684009,684021,684023,684027,684047,684069,
684081,684201,684229,684267,684427,684461,684483,684487,684603,684627,684661,
684663,684687,684803,684807,684829,684841,684883,684889,686003,686027,686047,
686061,686063,686067,686089,686201,686243,686281,686403,686407,686441,686489,
686621,686623,686649,686667,686683,686803,686809,686827,686841,686883,686887,
688029,688061,688081,688087,688207,688241,688249,688269,688289,688403,688443,
688449,688461,688467,688487,688603,688629,688641,688663,688667,688669,688687,
688821,688843,688889,800021,800023,800027,800029,800209,800221,800249,800263,
800283,800407,800423,800447,800469,800489,800603,800627,800643,800661,800681,
800801,800829,800841,800867,800887,802043,802047,802063,802067,802201,802207,
802283,802287,802403,802409,802421,802427,802601,802607,802623,802629,802803,
802809,802883,802887,804001,804009,804061,804069,804083,804087,804203,804207,
804221,804223,804287,804289,804441,804449,804461,804467,804487,804489,804641,
804649,804663,804669,804681,804683,804803,804807,804827,804829,804881,804883,
806003,806007,806063,806067,806081,806089,806243,806247,806267,806269,806283,
806289,806401,806409,806423,806429,806481,806487,806601,806609,806621,806627,
806683,806689,806843,806847,806861,806863,806881,806887,808041,808049,808061,
808069,808201,808203,808227,808229,808401,808403,808481,808489,808607,808609,
808681,808689,808807,808809,808821,808823,820003,820027,820083,820223,820229,
820247,820261,820409,820421,820423,820449,820481,820601,820607,820621,820641,

53


820647,820669,820801,820823,820827,820829,820849,820867,820887,822007,822023,
822041,822063,822083,822087,822089,822209,822227,822249,822403,822407,822429,
822441,822621,822647,822667,822669,822683,822801,822807,822829,822843,822847,
822863,824003,824009,824023,824043,824201,824221,824229,824243,824287,824421,
824427,824447,824463,824469,824487,824603,824629,824649,824661,824663,824667,
824689,824843,824869,824889,826043,826049,826061,826089,826203,826221,826229,
826241,826281,826407,826421,826427,826443,826483,826489,826607,826629,826643,
826661,826681,826683,826687,826827,826841,826869,828001,828023,828049,828063,
828067,828069,828081,828209,828261,828283,828443,828447,828463,828489,828607,
828643,828663,828687,828689,828803,828807,828829,828841,828847,828861,840001,
840029,840081,840203,840221,840227,840243,840287,840407,840421,840423,840429,
840443,840469,840489,840621,840623,840649,840667,840807,840809,840827,840847,
840849,840863,842007,842021,842043,842061,842063,842069,842087,842241,842249,
842261,842283,842401,842409,842423,842447,842449,842467,842603,842667,842681,
842809,842841,842861,842883,842889,844041,844043,844067,844083,844209,844227,
844229,844241,844281,844283,844429,844447,844463,844601,844623,844627,844647,
844687,844809,844823,844841,844867,844881,844887,844889,846001,846003,846021,
846041,846227,846229,846249,846261,846263,846289,846441,846463,846483,846607,
846623,846627,846641,846689,846801,846823,846843,846861,846867,846869,846883,
848009,848021,848047,848061,848081,848083,848089,848201,848209,848223,848247,
848407,848409,848423,848441,848449,848461,848603,848629,848643,848827,848849,
848863,848869,848881,860009,860021,860089,860201,860203,860223,860241,860243,
860267,860427,860429,860441,860463,860603,860621,860627,860629,860647,860661,
860681,860807,860823,860829,860847,860883,862003,862029,862047,862061,862067,
862069,862083,862201,862249,862269,862281,862287,862407,862463,862489,862601,
862609,862627,862641,862643,862663,862841,862849,862869,862887,864047,864049,
864063,864087,864201,864227,864249,864263,864281,864283,864289,864409,864423,
864427,864443,864483,864621,864643,864667,864801,864821,864823,864849,864887,
864889,866007,866009,866029,866049,866209,866227,866247,866261,866263,866269,
866287,866403,866423,866427,866449,866481,866649,866667,866687,866821,866823,
866841,866867,866869,866881,868001,868029,868043,868069,868081,868087,868089,
868223,868241,868261,868267,868289,868407,868421,868447,868601,868603,868627,
868641,868649,868669,868801,868809,868827,868843,880007,880023,880087,880209,
880221,880223,880227,880241,880263,880283,880403,880409,880429,880443,880449,
880461,880601,880627,880629,880641,880689,880821,880827,880843,880869,882003,
882027,882049,882067,882081,882083,882087,882203,882209,882221,882243,882247,
882267,882429,882443,882461,882463,882487,882603,882607,882621,882649,882801,
882823,882841,884001,884007,884027,884047,884247,884261,884281,884407,884421,
884441,884463,884467,884469,884481,884623,884629,884643,884661,884667,884683,
884809,884821,884829,884847,884883,886041,886047,886069,886081,886223,886249,

54


886261,886403,886421,886447,886469,886483,886487,886489,886603,886623,886629,
886647,886681,886687,886807,886821,886829,886849,886889,888009,888027,888041,
888061,888063,888067,888089,888203,888207,888221,888243,888249,888269,888403,
888447,888467,888481,888483,888643,888647,888667,888681,888801,888869,888887}
;

55


CONCLUSIÓN

En éste trabajo podemos notar que el campo de la Matemática no tiene límites, además de
ser la madre de todas las ciencias es aquella que nunca te deja de enseñar cosas nuevas y
ejemplo es el trabajo realizado sobre la Aritmética Sin Llevar, la que además de aprender
en qué consistían sus operaciones nos dimos cuenta que es un anillo unitario y
conmutativo. Esta Aritmética goza de un rico potencial matemático. Además notamos el
extraño comportamiento que presentan los números primos en ésta nueva aritmética así
como también los números pares.
Ha sido muy importante éste trabajo ya que como matemática me siento contenta en
saber que la carrera que escogí estudiar no tiene límites en cuanto a enseñanza se trata.

56


RECOMENDACIONES

 Recomendaría que se estudiase también:
¿Qué pasa con la factorización de números en el producto de primos Sin Llevar?
Por desgracia, la existencia de divisores de cero complica las cosas, y resulta que
no hay manera natural de definir ésta forma de factorizar. Recomiendo que se
haga un estudio sobre ésta problemática en la Aritmética Sin Llevar.
 Impulsar el desarrollo de ésta nueva Aritmética de manera que se pueda continuar
haciendo Matemática.

57


BIBLIOGRAFÍA.

1. DAVID APPLEGATE, MARC LEBRUN Y NEIL SLOANE. Julio 7, 2011.
Carryless Arithmetic Mod 10.

58

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