La Conjetura de Goldbach en la Aritmética Dismal

LA CONJETURA DE GOLDBACH EN LA
ARITMÉTICA DISMAL

José Rosales

VI [OTRAS ARITMÉTICAS]

138

[LA CONJETURA DE GOLDBACH EN LA ARITMÉTICA DISMAL] VI

AGRADECIMIENTOS

A DIOS TODOPODEROSO:
Por cuidarme y bendecirme siempre en todos mis estudios y por haberme permitido la
culminación de esta carrera.
A MI ESPOSA Y A MI HIJO:
Luris Jaén y Joseph Rosales por ser mi fuente de inspiración.
A MIS PADRES:
Israel Pérez y Eulogia Rosales.
Al Doctor JAIME GUTIÉRREZ por haberme guiado y ayudado durante el semestre.
A MIS COMPAÑEROS DE GRUPO: Los cuales siempre aportaron su grano de arena
para la realización de este trabajo.

139


DEDICATORIA

A ti mi Divino Dios que me ha dado la vida y fortaleza para terminar este trabajo de
Graduación.

A mi esposa Luris por apoyarme y ayudarme en los
momentos más difíciles.

A mi hijo Joseph por ser fuente de inspiración.

A mis Padres por brindarme toda su ayuda.

140


INTRODUCCION

En matemática, una conjetura es una afirmación que se supone cierta, pero que carece de
demostración. Ejemplo de ello es la famosa conjetura conocida como la conjetura de
Goldbach, la cual nos dice “Todo número par mayor que dos puede ser escrito como
suma de dos números primos”.

El Objetivo principal de este trabajo es de realizar un estudio sobre la conjetura de
Goldbach pero bajo los parámetros de una nueva Aritmética, “La Aritmética Dismal.

En la aritmética tradicional se cree que ésta conjetura es cierta. Pero ¿Será cierta la
conjetura de Goldbach en la aritmética dismal? Para saber si es o no cierta primero
debemos estudiar cada una de las propiedades con respecto a ésta nueva aritmética.

Espero que éste trabajo sea de su completo agrado.

141


Historia sobre el enunciado de la Conjetura de Goldbach

La verdad histórica sobre la enunciación de la conjetura de Goldbach
En 1742 Euler se había trasladado a Berlín y Goldbach le escribe a su amigo sobre
nuevas proposiciones que ha concebido relacionadas con los números primos:
[…] quisiera aventurar una conjetura: todo número que esté formado por dos números
primos es una suma de tantos números primos como se desee (contando entre ellos a las
unidades), hasta alcanzar solo unidades.
Pero su especulación no se detiene allí. Al leer lo ya
escrito reconoce que pudiera mejorar su conjetura y
escribe al margen:
[...] Al volver a leer esto encuentro que esta
conjetura se pudiera demostrar con sumo rigor en
el caso n+1, si se cumple en el caso n y n+1 se
divide en dos números primos.
Euler envía como respuesta a la consideración
siguiente:
Que un número que sea resoluble en dos números primos, se puede descomponer a la vez
en tantos números primos como se quiera, puede ser ilustrado y confirmado a partir de
una observación que su excelencia me había comunicado anteriormente, que todo
número par es una suma de dos números primos.
Entonces Euler añade:
Pero el que todo número par sea una suma de dos primos lo considero un teorema, a
pesar de que no puedo demostrarlo…
Sin dudas Euler aplicó su gran ingenio para tratar de demostrar lo que el considera un
teorema:
Todo número par mayor que 2 puede ser escrito al menos de una forma como suma de
dos números primos.

142


Resultados que apoyan la Conjetura de Goldbach

Hay resultados que han aportado grandes matemáticos a lo largo de la historia que
aportan un sin número de evidencias que apuntan a que esta conjetura sea verdadera.
Tenemos grandes resultados Teóricos, evidencias Numéricas, evidencias Probabilísticas y
También hay diversas demostraciones que todavía no han tenido el visto bueno de que
estén demostradas correctamente. He aquí algo de lo antes mencionado.

Las evidencias numéricas y todos los resultados teóricos apuntan a la
veracidad de la Conjetura de Goldbach.
Existen grandes resultados de Matemáticos
Resultados Teóricos entre ellos: Schnirelman, Vinogradov,
Deshouillers , Ramaré, entre otros.

- En 1855, A. Desboves verificó la conjetura
Evidencias de Goldbach para n <=10.000.
Numéricas - En 1940, N. Pipping verificó la conjetura
hasta 100.000.

Cuanto mayor es el número más
Evidencias posibilidades existen de descomponerlo en
Probabilísticas sumandos primos.

También existen diversas demostraciones de la conjetura de Goldbach, no
obstante, luego de más de 200 años no se ha reconocido ninguna de las
muchas supuestas pruebas como válida.

143


Operaciones en la Aritmética Dismal

Suma Dismal de un dígito:

Dados dos números m y n de una sola cifra se define m + n = max {m, n}.

Ejemplos: 1) 3 + 4= máx {3, 4}= 4

2) 5 + 8= máx {5, 8}= 8

Suma Dismal de n dígitos:

Ejemplos: 1) 3145 + 4653= máx {{3, 4}, {1,6}, {4,5}, {5,3}}= 4655

2) 1368 + 7931= max{{1,7}, {3,9}, {6, 3}, {8,1}}= 7968

Producto Dismal:

Dados dos números m y n de una sola cifra se define m*n = min {m, n}.

Ejemplos: 1) 2*6= 6

2) 7419*354= 354454

A continuación vamos a ilustrar los ejemplos dados con los respectivos programas hechos
en Matemática 8.0 que calculan la suma y el producto dismal.

144


Pares en la Aritmética Dismal

Definición: Un número entero m es un número par dismal si y solo si existe otro número
entero tal que m=19*n

19 es el análogo a 2 por ende 19 es el primo par dismal más pequeño. Debido a ello los
múltiplos dismales de 19 son una opción para definir los números pares dismales.

Los pares dismales son todos los múltiplos de 19.

Ejemplos: Los primeros 99 pares dismales son los siguientes:

11 12 13 14 15 16 17 18 19 110
111 112 113 114 115 116 117 118 119 120
121 122 123 124 125 126 127 128 129 130
131 132 133 134 135 136 137 138 139 140
141 142 143 144 145 146 147 148 149 150
151 152 153 154 155 156 157 158 159 160
161 162 163 164 165 166 167 168 169 170
171 172 173 174 175 176 177 178 179 180
181 182 183 184 185 186 187 188 189 190
191 192 193 194 195 196 197 198 199

Números de Goldbach

Definición: Un número n es un número de Goldbach dismal si es Par Dismal, 9-Digital y
no termina en cero.

Los números de Goldbach son los múltiplos de 19 que podrían escribirse como suma de
dos números primos.

Los números que cumplen éstas condiciones pueden ser expresados o escritos como suma
de dos primos dismales.

145


Conjetura Dismal de Goldbach

"Todo número de Goldbach dismal es suma dismal de dos primos dismales"

Además los sumandos presentan las siguientes características:

-Primo Principal: Es el primo con igual longitud que el número par dismal dado y su
primera cifra es 1, en el caso de que los sumandos tengan la misma longitud el primo
principal será el mayor primo dismal.

-Primo Secundario: Es el sumando de menor longitud.

146


Programas y Datos Utilizados

Condiciones para que un par dismal pueda ser escrito como suma de dos primos
dismales.

- Debe tener un 9 en alguna de sus cifras.
- No puede terminar en cero salvo 90.

Esta función elimina en un determinado intervalo los números que no cumplan las
condiciones mencionadas para cualquier lista de números enteros.

147


Primos Dismales

Utilizaremos los primos dismales de 2, 3, 4 y 5 cifras.

PD2C={19,29,39,49,59,69,79,89,90,91,92, 93, 94, 95, 96, 97, 98, 99};

PD3C={109,209,219,309,319,329,409,419,429,439,509,519,529,539,549,609,619,629,6
39,649,659,709,719,729,739,749,759,769,809,819,829,839,849,859,869,879,901,902,903
,904,905,906,907,908,909,912,913,914,915,916,917,918,919,923,924,925,926,927,928,9
29,934,935,936,937,938,939,945,946,947,948,949,956,957,958,959,967,968,969,978,979
,989};
PD4C={1009,1019,1029,1039,1049,1059,1069,1079,1089,1091,1092,1093,1094,1095,1
096,1097,1098,1099,1109,1209,1309,1409,1509,1609,1709,1809,1901,1902,1903,1904,1
905,1906,1907,1908,1909};

PD5C={10009,10019,10029,10039,10049,10059,10069,10079,10089,10091,10092,1009
3,10094,10095,10096,10097,10098,10099,10119,10129,10139,10149,10159,10169,1017
9,10189,10191,10192,10193,10194,10195,10196,10197,10198,10199,10219,10229,1023
9,10249,10259,10269,10279,10289,10291,10292,10293,10294,10295,10296,10297,1029
8,10299,10319,10329,10339,10349,10359,10369,10379,10389,10391,10392,10393,1039
4,10395,10396,10397,10398,10399,10419,10429,10439,10449,10459,10469,10479,1048
9,10491,10492,10493,10494,10495,10496,10497,10498,10499,10519,10529,10539,1054
9,10559,10569,10579,10589,10591,10592,10593,10594,10595,10596,10597,10598,1059
9,10619,10629,10639,10649,10659,10669,10679,10689,10691,10692,10693,10694,1069
5,10696,10697,10698,10699,10719,10729,10739,10749,10759,10769,10779,10789,1079
1,10792,10793,10794,10795,10796,10797,10798,10799,10819,10829,10839,10849,1085
9,10869,10879,10889,10891,10892,10893,10894,10895,10896,10897,10898,10899,1091
1,10912,10913,10914,10915,10916,10917,10918,10919,10921,10922,10923,10924,1092
5,10926,10927,10928,10929,10931,10932,10933,10934,10935,10936,10937,10938,1093
9,10941,10942,10943,10944,10945,10946,10947,10948,10949,10951,10952,10953,1095
4,10955,10956,10957,10958,10959,10961,10962,10963,10964,10965,10966,10967,1096

148


8,10969,10971,10972,10973,10974,10975,10976,10977,10978,10979,10981,10982,1098
3,10984,10985,10986,10987,10988,10989,10991,10992,10993,10994,10995,10996,1099
7,10998,10999,11009,11109,11209,11309,11409,11509,11609,11709,11809,11901,1190
2,11903,11904,11905,11906,11907,11908,11909,12009,12029,12039,12049,12059,1206
9,12079,12089,12091,12092,12093,12094,12095,12096,12097,12098,12099,12109,1220
9,12309,12409,12509,12609,12709,12809,12901,12902,12903,12904,12905,12906,1290
7,12908,12909,13009,13029,13039,13049,13059,13069,13079,13089,13091,13092,1309
3,13094,13095,13096,13097,13098,13099,13109,13209,13309,13409,13509,13609,1370
9,13809,13901,13902,13903,13904,13905,13906,13907,13908,13909,14009,14029,1403
9,14049,14059,14069,14079,14089,14091,14092,14093,14094,14095,14096,14097,1409
8,14099,14109,14209,14309,14409,14509,14609,14709,14809,14901,14902,14903,1490
4,14905,14906,14907,14908,14909,15009,15029,15039,15049,15059,15069,15079,1508
9,15091,15092,15093,15094,15095,15096,15097,15098,15099,15109,15209,15309,1540
9,15509,15609,15709,15809,15901,15902,15903,15904,15905,15906,15907,15908,1590
9,16009,16029,16039,16049,16059,16069,16079,16089,16091,16092,16093,16094,1609
5,16096,16097,16098,16099,16109,16209,16309,16409,16509,16609,16709,16809,1690
1,16902,16903,16904,16905,16906,16907,16908,16909,17009,17029,17039,17049,1705
9,17069,17079,17089,17091,17092,17093,17094,17095,17096,17097,17098,17099,1710
9,17209,17309,17409,17509,17609,17709,17809,17901,17902,17903,17904,17905,1790
6,17907,17908,17909,18009,18029,18039,18049,18059,18069,18079,18089,18091,1809
2,18093,18094,18095,18096,18097,18098,18099,18109,18209,18309,18409,18509,1860
9,18709,18809,18901,18902,18903,18904,18905,18906,18907,18908,18909,19001,1900
2,19003,19004,19005,19006,19007,19008,19009,19021,19022,19023,19024,19025,1902
6,19027,19028,19029,19031,19032,19033,19034,19035,19036,19037,19038,19039,1904
1,19042,19043,19044,19045,19046,19047,19048,19049,19051,19052,19053,19054,1905
5,19056,19057,19058,19059,19061,19062,19063,19064,19065,19066,19067,19068,1906
9,19071,19072,19073,19074,19075,19076,19077,19078,19079,19081,19082,19083,1908
4,19085,19086,19087,19088,19089,19091,19092,19093,19094,19095,19096,19097,1909
8,19099,19101,19102,19103,19104,19105,19106,19107,19108,19109,19201,19202,1920
3,19204,19205,19206,19207,19208,19209,19301,19302,19303,19304,19305,19306,1930
7,19308,19309,19401,19402,19403,19404,19405,19406,19407,19408,19409,19501,1950
2,19503,19504,19505,19506,19507,19508,19509,19601,19602,19603,19604,19605,1960
6,19607,19608,19609,19701,19702,19703,19704,19705,19706,19707,19708,19709,1980
1,19802,19803,19804,19805,19806,19807,19808,19809,19901,19902,19903,19904,1990
5,19906,19907,19908,19909};

149


Suma dismal entre primos dismales de tres cifras

1) A continuación haremos la suma dismal de los primos dismales de 2 cifras con 109
que es el único primo dismal de 3 cifras que empieza con 1.
2) Haremos la intersección de éstos números con H[A], para encontrar los números de 3
cifras que se pueden expresar como suma de dos primos dismales.

Suma dismal entre primos dismales de 4, 3 y 2 cifras.

Para el caso de 4 cifras, hay que sumar todos los primos de 4 cifras que empiezan con 1
entre ellos, luego todos los primos de 4 cifras que empiezan con 1 con todos los primos
de 2 y 3 cifras.
A continuación haremos la suma dismal entre los primos de 4 cifras.

{1009,1019,1029,1039,1049,1059,1069,1079,1089,1099,1091,1092,1093,1094,1095,109
6,1097,1098,1109,1119,1129,1139,1149,1159,1169,1179,1189,1199,1209,1219,1229,123
9,1249,1259,1269,1279,1289,1299,1309,1319,1329,1339,1349,1359,1369,1379,1389,139
9,1409,1419,1429,1439,1449,1459,1469,1479,1489,1499,1509,1519,1529,1539,1549,155
9,1569,1579,1589,1599,1609,1619,1629,1639,1649,1659,1669,1679,1689,1699,1709,171
9,1729,1739,1749,1759,1769,1779,1789,1799,1809,1819,1829,1839,1849,1859,1869,187
9,1889,1899,1909,1919,1929,1939,1949,1959,1969,1979,1989,1991,1992,1993,1994,199
5,1996,1997,1998,1999,1901,1902,1903,1904,1905,1906,1907,1908}

{1119,1129,1139,1149,1159,1169,1179,1189,1199,1219,1229,1239,1249,1259,1269,127
9,1289,1299,1319,1329,1339,1349,1359,1369,1379,1389,1399,1419,1429,1439,1449,145

150


9,1469,1479,1489,1499,1519,1529,1539,1549,1559,1569,1579,1589,1599,1619,1629,163
9,1649,1659,1669,1679,1689,1699,1719,1729,1739,1749,1759,1769,1779,1789,1799,181
9,1829,1839,1849,1859,1869,1879,1889,1899,1919,1929,1939,1949,1959,1969,1979,198
9,1991,1992,1993,1994,1995,1996,1997,1998,1999}

Suma dismal entre primos primarios de 4 cifras con secundarios de 3 cifras.

{1109,1119,1129,1139,1149,1159,1169,1179,1189,1199,1209,1309,1409,1509,1609,170
9,1809,1909,1219,1229,1239,1249,1259,1269,1279,1289,1299,1319,1419,1519,1619,171
9,1819,1919,1329,1339,1349,1359,1369,1379,1389,1399,1429,1529,1629,1729,1829,192
9,1439,1449,1459,1469,1479,1489,1499,1539,1639,1739,1839,1939,1549,1559,1569,157
9,1589,1599,1649,1749,1849,1949,1659,1669,1679,1689,1699,1759,1859,1959,1769,177
9,1789,1799,1869,1969,1879,1889,1899,1979,1989,1991,1992,1993,1994,1995,1996,199
7,1998,1999,1901,1902,1903,1904,1905,1906,1907,1908,1912,1913,1914,1915,1916,191
7,1918,1923,1924,1925,1926,1927,1928,1934,1935,1936,1937,1938,1945,1946,1947,194
8,1956,1957,1958,1967,1968,1978}

{1119,1129,1139,1149,1159,1169,1179,1189,1199,1219,1229,1239,1249,1259,1269,127
9,1289,1299,1319,1329,1339,1349,1359,1369,1379,1389,1399,1419,1429,1439,1449,145
9,1469,1479,1489,1499,1519,1529,1539,1549,1559,1569,1579,1589,1599,1619,1629,163
9,1649,1659,1669,1679,1689,1699,1719,1729,1739,1749,1759,1769,1779,1789,1799,181
9,1829,1839,1849,1859,1869,1879,1889,1899,1912,1913,1914,1915,1916,1917,1918,191
9,1923,1924,1925,1926,1927,1928,1929,1934,1935,1936,1937,1938,1939,1945,1946,194
7,1948,1949,1956,1957,1958,1959,1967,1968,1969,1978,1979,1989,1991,1992,1993,199
4,1995,1996,1997,1998,1999}

151


Suma dismal entre primos primarios de 4 cifras con secundarios de 2 cifras.

{1019,1029,1039,1049,1059,1069,1079,1089,1099,1119,1219,1319,1419,1519,1619,171
9,1819,1919,1129,1229,1329,1429,1529,1629,1729,1829,1929,1139,1239,1339,1439,153
9,1639,1739,1839,1939,1149,1249,1349,1449,1549,1649,1749,1849,1949,1159,1259,135
9,1459,1559,1659,1759,1859,1959,1169,1269,1369,1469,1569,1669,1769,1869,1969,117
9,1279,1379,1479,1579,1679,1779,1879,1979,1189,1289,1389,1489,1589,1689,1789,188
9,1989,1091,1092,1093,1094,1095,1096,1097,1098,1199,1299,1399,1499,1599,1699,179
9,1899,1991,1992,1993,1994,1995,1996,1997,1998,1999}

{1119,1129,1139,1149,1159,1169,1179,1189,1199,1219,1229,1239,1249,1259,1269,127
9,1289,1299,1319,1329,1339,1349,1359,1369,1379,1389,1399,1419,1429,1439,1449,145
9,1469,1479,1489,1499,1519,1529,1539,1549,1559,1569,1579,1589,1599,1619,1629,163
9,1649,1659,1669,1679,1689,1699,1719,1729,1739,1749,1759,1769,1779,1789,1799,181
9,1829,1839,1849,1859,1869,1879,1889,1899,1912,1913,1914,1915,1916,1917,1918,191
9,1923,1924,1925,1926,1927,1928,1929,1934,1935,1936,1937,1938,1939,1945,1946,194
7,1948,1949,1956,1957,1958,1959,1967,1968,1969,1978,1979,1989,1991,1992,1993,199
4,1995,1996,1997,1998,1999}

Números de 4 cifras que se pueden expresar como suma de dos primos dismales

La unión de C1, C2 y C3 nos dará los números de 4 cifras que se pueden expresar como
suma de dos primos dismales.

{1119,1129,1139,1149,1159,1169,1179,1189,1199,1219,1229,1239,1249,1259,1269,127
9,1289,1299,1319,1329,1339,1349,1359,1369,1379,1389,1399,1419,1429,1439,1449,145
9,1469,1479,1489,1499,1519,1529,1539,1549,1559,1569,1579,1589,1599,1619,1629,163

152


9,1649,1659,1669,1679,1689,1699,1719,1729,1739,1749,1759,1769,1779,1789,1799,181
9,1829,1839,1849,1859,1869,1879,1889,1899,1912,1913,1914,1915,1916,1917,1918,191
9,1923,1924,1925,1926,1927,1928,1929,1934,1935,1936,1937,1938,1939,1945,1946,194
7,1948,1949,1956,1957,1958,1959,1967,1968,1969,1978,1979,1989,1991,1992,1993,199
4,1995,1996,1997,1998,1999}´

Números de 4 cifras que no se pueden expresar como suma de dos primos dismales

{1191,1192,1193,1194,1195,1196,1197,1198,1291,1292,1293,1294,1295,1296,1297,129
8,1391,1392,1393,1394,1395,1396,1397,1398,1491,1492,1493,1494,1495,1496,1497,149
8,1591,1592,1593,1594,1595,1596,1597,1598,1691,1692,1693,1694,1695,1696,1697,169
8,1791,1792,1793,1794,1795,1796,1797,1798,1891,1892,1893,1894,1895,1896,1897,189
8,1911,1921,1922,1931,1932,1933,1941,1942,1943,1944,1951,1952,1953,1954,1955,196
1,1962,1963,1964,1965,1966,1971,1972,1973,1974,1975,1976,1977,1981,1982,1983,198
4,1985,1986,1987,1988}

Suma entre primos dismales de 5, 4, 3 y 2 cifras

Para el caso de 5 cifras se suman todos los primos de 5 cifras que empiezan con 1 entre
ellos y luego se suman los primos de 5 cifras que empiezan con 1 con todos los primos de
4, 3 y 2 cifras.

{10009,10019,10029,10039,10049,10059,10069,10079,10089,10099,10091,10092,10093
,10094,10095,10096,10097,10098,10119,10129,10139,10149,10159,10169,10179,10189,
10199,10191,10192,10193,10194,10195,10196,10197,10198,10219,10229,10239,10249,
10259,10269,10279,10289,10299,10291,10292,10293,10294,10295,10296,10297,10298,
10319,10329,10339,10349,10359,10369,10379,10389,10399,10391,10392,10393,10394,
10395,10396,10397,10398,10419,10429,10439,10449,10459,10469,10479,10489,10499,
10491,10492,10493,10494,10495,10496,10497,10498,10519,10529,10539,10549,10559,
10569,10579,10589,10599,10591,10592,10593,10594,10595,10596,10597,10598,10619,

153


10629,10639,10649,10659,10669,10679,10689,10699,10691,10692,10693,10694,10695,
10696,10697,10698,10719,10729,10739,10749,10759,10769,10779,10789,10799,10791,
10792,10793,10794,10795,10796,10797,10798,10819,10829,10839,10849,10859,10869,
10879,10889,10899,10891,10892,10893,10894,10895,10896,10897,10898,10919,10929,
10939,10949,10959,10969,10979,10989,10991,10992,10993,10994,10995,10996,10997,
10998,10999,10911,10912,10913,10914,10915,10916,10917,10918,10921,10922,10923,
10924,10925,10926,10927,10928,10931,10932,10933,10934,10935,10936,10937,10938,
10941,10942,10943,10944,10945,10946,10947,10948,10951,10952,10953,10954,10955,
10956,10957,10958,10961,10962,10963,10964,10965,10966,10967,10968,10971,10972,
10973,10974,10975,10976,10977,10978,10981,10982,10983,10984,10985,10986,10987,
10988,11009,11019,11029,11039,11049,11059,11069,11079,11089,11099,11119,11129,
11139,11149,11159,11169,11179,11189,11199,11219,11229,11239,11249,11259,11269,
11279,11289,11299,11319,11329,11339,11349,11359,11369,11379,11389,11399,11419,
11429,11439,11449,11459,11469,11479,11489,11499,11519,11529,11539,11549,11559,
11569,11579,11589,11599,11619,11629,11639,11649,11659,11669,11679,11689,11699,
11719,11729,11739,11749,11759,11769,11779,11789,11799,11819,11829,11839,11849,
11859,11869,11879,11889,11899,11919,11929,11939,11949,11959,11969,11979,11989,
11999,11109,11209,11309,11409,11509,11609,11709,11809,11909,11991,11992,11993,
11994,11995,11996,11997,11998,11911,11912,11913,11914,11915,11916,11917,11918,
11921,11922,11923,11924,11925,11926,11927,11928,11931,11932,11933,11934,11935,
11936,11937,11938,11941,11942,11943,11944,11945,11946,11947,11948,11951,11952,
11953,11954,11955,11956,11957,11958,11961,11962,11963,11964,11965,11966,11967,
11968,11971,11972,11973,11974,11975,11976,11977,11978,11981,11982,11983,11984,
11985,11986,11987,11988,11901,11902,11903,11904,11905,11906,11907,11908,12009,
12019,12029,12039,12049,12059,12069,12079,12089,12099,12119,12129,12139,12149,
12159,12169,12179,12189,12199,12219,12229,12239,12249,12259,12269,12279,12289,
12299,12319,12329,12339,12349,12359,12369,12379,12389,12399,12419,12429,12439,
12449,12459,12469,12479,12489,12499,12519,12529,12539,12549,12559,12569,12579,
12589,12599,12619,12629,12639,12649,12659,12669,12679,12689,12699,12719,12729,
12739,12749,12759,12769,12779,12789,12799,12819,12829,12839,12849,12859,12869,
12879,12889,12899,12919,12929,12939,12949,12959,12969,12979,12989,12999,12109,
12209,12309,12409,12509,12609,12709,12809,12909,12091,12092,12093,12094,12095,
12096,12097,12098,12191,12192,12193,12194,12195,12196,12197,12198,12291,12292,
12293,12294,12295,12296,12297,12298,12391,12392,12393,12394,12395,12396,12397,
12398,12491,12492,12493,12494,12495,12496,12497,12498,12591,12592,12593,12594,
12595,12596,12597,12598,12691,12692,12693,12694,12695,12696,12697,12698,12791,
12792,12793,12794,12795,12796,12797,12798,12891,12892,12893,12894,12895,12896,
12897,12898,12991,12992,12993,12994,12995,12996,12997,12998,12911,12912,12913,
12914,12915,12916,12917,12918,12921,12922,12923,12924,12925,12926,12927,12928,
12931,12932,12933,12934,12935,12936,12937,12938,12941,12942,12943,12944,12945,
12946,12947,12948,12951,12952,12953,12954,12955,12956,12957,12958,12961,12962,
12963,12964,12965,12966,12967,12968,12971,12972,12973,12974,12975,12976,12977,
12978,12981,12982,12983,12984,12985,12986,12987,12988,12901,12902,12903,12904,
12905,12906,12907,12908,13009,13019,13029,13039,13049,13059,13069,13079,13089,
13099,13119,13129,13139,13149,13159,13169,13179,13189,13199,13219,13229,13239,
13249,13259,13269,13279,13289,13299,13319,13329,13339,13349,13359,13369,13379,

154


13389,13399,13419,13429,13439,13449,13459,13469,13479,13489,13499,13519,13529,
13539,13549,13559,13569,13579,13589,13599,13619,13629,13639,13649,13659,13669,
13679,13689,13699,13719,13729,13739,13749,13759,13769,13779,13789,13799,13819,
13829,13839,13849,13859,13869,13879,13889,13899,13919,13929,13939,13949,13959,
13969,13979,13989,13999,13109,13209,13309,13409,13509,13609,13709,13809,13909,
13091,13092,13093,13094,13095,13096,13097,13098,13191,13192,13193,13194,13195,
13196,13197,13198,13291,13292,13293,13294,13295,13296,13297,13298,13391,13392,
13393,13394,13395,13396,13397,13398,13491,13492,13493,13494,13495,13496,13497,
13498,13591,13592,13593,13594,13595,13596,13597,13598,13691,13692,13693,13694,
13695,13696,13697,13698,13791,13792,13793,13794,13795,13796,13797,13798,13891,
13892,13893,13894,13895,13896,13897,13898,13991,13992,13993,13994,13995,13996,
13997,13998,13911,13912,13913,13914,13915,13916,13917,13918,13921,13922,13923,
13924,13925,13926,13927,13928,13931,13932,13933,13934,13935,13936,13937,13938,
13941,13942,13943,13944,13945,13946,13947,13948,13951,13952,13953,13954,13955,
13956,13957,13958,13961,13962,13963,13964,13965,13966,13967,13968,13971,13972,
13973,13974,13975,13976,13977,13978,13981,13982,13983,13984,13985,13986,13987,
13988,13901,13902,13903,13904,13905,13906,13907,13908,14009,14019,14029,14039,
14049,14059,14069,14079,14089,14099,14119,14129,14139,14149,14159,14169,14179,
14189,14199,14219,14229,14239,14249,14259,14269,14279,14289,14299,14319,14329,
14339,14349,14359,14369,14379,14389,14399,14419,14429,14439,14449,14459,14469,
14479,14489,14499,14519,14529,14539,14549,14559,14569,14579,14589,14599,14619,
14629,14639,14649,14659,14669,14679,14689,14699,14719,14729,14739,14749,14759,
14769,14779,14789,14799,14819,14829,14839,14849,14859,14869,14879,14889,14899,
14919,14929,14939,14949,14959,14969,14979,14989,14999,14109,14209,14309,14409,
14509,14609,14709,14809,14909,14091,14092,14093,14094,14095,14096,14097,14098,
14191,14192,14193,14194,14195,14196,14197,14198,14291,14292,14293,14294,14295,
14296,14297,14298,14391,14392,14393,14394,14395,14396,14397,14398,14491,14492,
14493,14494,14495,14496,14497,14498,14591,14592,14593,14594,14595,14596,14597,
14598,14691,14692,14693,14694,14695,14696,14697,14698,14791,14792,14793,14794,
14795,14796,14797,14798,14891,14892,14893,14894,14895,14896,14897,14898,14991,
14992,14993,14994,14995,14996,14997,14998,14911,14912,14913,14914,14915,14916,
14917,14918,14921,14922,14923,14924,14925,14926,14927,14928,14931,14932,14933,
14934,14935,14936,14937,14938,14941,14942,14943,14944,14945,14946,14947,14948,
14951,14952,14953,14954,14955,14956,14957,14958,14961,14962,14963,14964,14965,
14966,14967,14968,14971,14972,14973,14974,14975,14976,14977,14978,14981,14982,
14983,14984,14985,14986,14987,14988,14901,14902,14903,14904,14905,14906,14907,
14908,15009,15019,15029,15039,15049,15059,15069,15079,15089,15099,15119,15129,
15139,15149,15159,15169,15179,15189,15199,15219,15229,15239,15249,15259,15269,
15279,15289,15299,15319,15329,15339,15349,15359,15369,15379,15389,15399,15419,
15429,15439,15449,15459,15469,15479,15489,15499,15519,15529,15539,15549,15559,
15569,15579,15589,15599,15619,15629,15639,15649,15659,15669,15679,15689,15699,
15719,15729,15739,15749,15759,15769,15779,15789,15799,15819,15829,15839,15849,
15859,15869,15879,15889,15899,15919,15929,15939,15949,15959,15969,15979,15989,
15999,15109,15209,15309,15409,15509,15609,15709,15809,15909,15091,15092,15093,
15094,15095,15096,15097,15098,15191,15192,15193,15194,15195,15196,15197,15198,
15291,15292,15293,15294,15295,15296,15297,15298,15391,15392,15393,15394,15395,

155


15396,15397,15398,15491,15492,15493,15494,15495,15496,15497,15498,15591,15592,
15593,15594,15595,15596,15597,15598,15691,15692,15693,15694,15695,15696,15697,
15698,15791,15792,15793,15794,15795,15796,15797,15798,15891,15892,15893,15894,
15895,15896,15897,15898,15991,15992,15993,15994,15995,15996,15997,15998,15911,
15912,15913,15914,15915,15916,15917,15918,15921,15922,15923,15924,15925,15926,
15927,15928,15931,15932,15933,15934,15935,15936,15937,15938,15941,15942,15943,
15944,15945,15946,15947,15948,15951,15952,15953,15954,15955,15956,15957,15958,
15961,15962,15963,15964,15965,15966,15967,15968,15971,15972,15973,15974,15975,
15976,15977,15978,15981,15982,15983,15984,15985,15986,15987,15988,15901,15902,
15903,15904,15905,15906,15907,15908,16009,16019,16029,16039,16049,16059,16069,
16079,16089,16099,16119,16129,16139,16149,16159,16169,16179,16189,16199,16219,
16229,16239,16249,16259,16269,16279,16289,16299,16319,16329,16339,16349,16359,
16369,16379,16389,16399,16419,16429,16439,16449,16459,16469,16479,16489,16499,
16519,16529,16539,16549,16559,16569,16579,16589,16599,16619,16629,16639,16649,
16659,16669,16679,16689,16699,16719,16729,16739,16749,16759,16769,16779,16789,
16799,16819,16829,16839,16849,16859,16869,16879,16889,16899,16919,16929,16939,
16949,16959,16969,16979,16989,16999,16109,16209,16309,16409,16509,16609,16709,
16809,16909,16091,16092,16093,16094,16095,16096,16097,16098,16191,16192,16193,
16194,16195,16196,16197,16198,16291,16292,16293,16294,16295,16296,16297,16298,
16391,16392,16393,16394,16395,16396,16397,16398,16491,16492,16493,16494,16495,
16496,16497,16498,16591,16592,16593,16594,16595,16596,16597,16598,16691,16692,
16693,16694,16695,16696,16697,16698,16791,16792,16793,16794,16795,16796,16797,
16798,16891,16892,16893,16894,16895,16896,16897,16898,16991,16992,16993,16994,
16995,16996,16997,16998,16911,16912,16913,16914,16915,16916,16917,16918,16921,
16922,16923,16924,16925,16926,16927,16928,16931,16932,16933,16934,16935,16936,
16937,16938,16941,16942,16943,16944,16945,16946,16947,16948,16951,16952,16953,
16954,16955,16956,16957,16958,16961,16962,16963,16964,16965,16966,16967,16968,
16971,16972,16973,16974,16975,16976,16977,16978,16981,16982,16983,16984,16985,
16986,16987,16988,16901,16902,16903,16904,16905,16906,16907,16908,17009,17019,
17029,17039,17049,17059,17069,17079,17089,17099,17119,17129,17139,17149,17159,
17169,17179,17189,17199,17219,17229,17239,17249,17259,17269,17279,17289,17299,
17319,17329,17339,17349,17359,17369,17379,17389,17399,17419,17429,17439,17449,
17459,17469,17479,17489,17499,17519,17529,17539,17549,17559,17569,17579,17589,
17599,17619,17629,17639,17649,17659,17669,17679,17689,17699,17719,17729,17739,
17749,17759,17769,17779,17789,17799,17819,17829,17839,17849,17859,17869,17879,
17889,17899,17919,17929,17939,17949,17959,17969,17979,17989,17999,17109,17209,
17309,17409,17509,17609,17709,17809,17909,17091,17092,17093,17094,17095,17096,
17097,17098,17191,17192,17193,17194,17195,17196,17197,17198,17291,17292,17293,
17294,17295,17296,17297,17298,17391,17392,17393,17394,17395,17396,17397,17398,
17491,17492,17493,17494,17495,17496,17497,17498,17591,17592,17593,17594,17595,
17596,17597,17598,17691,17692,17693,17694,17695,17696,17697,17698,17791,17792,
17793,17794,17795,17796,17797,17798,17891,17892,17893,17894,17895,17896,17897,
17898,17991,17992,17993,17994,17995,17996,17997,17998,17911,17912,17913,17914,
17915,17916,17917,17918,17921,17922,17923,17924,17925,17926,17927,17928,17931,
17932,17933,17934,17935,17936,17937,17938,17941,17942,17943,17944,17945,17946,
17947,17948,17951,17952,17953,17954,17955,17956,17957,17958,17961,17962,17963,

156


17964,17965,17966,17967,17968,17971,17972,17973,17974,17975,17976,17977,17978,
17981,17982,17983,17984,17985,17986,17987,17988,17901,17902,17903,17904,17905,
17906,17907,17908,18009,18019,18029,18039,18049,18059,18069,18079,18089,18099,
18119,18129,18139,18149,18159,18169,18179,18189,18199,18219,18229,18239,18249,
18259,18269,18279,18289,18299,18319,18329,18339,18349,18359,18369,18379,18389,
18399,18419,18429,18439,18449,18459,18469,18479,18489,18499,18519,18529,18539,
18549,18559,18569,18579,18589,18599,18619,18629,18639,18649,18659,18669,18679,
18689,18699,18719,18729,18739,18749,18759,18769,18779,18789,18799,18819,18829,
18839,18849,18859,18869,18879,18889,18899,18919,18929,18939,18949,18959,18969,
18979,18989,18999,18109,18209,18309,18409,18509,18609,18709,18809,18909,18091,
18092,18093,18094,18095,18096,18097,18098,18191,18192,18193,18194,18195,18196,
18197,18198,18291,18292,18293,18294,18295,18296,18297,18298,18391,18392,18393,
18394,18395,18396,18397,18398,18491,18492,18493,18494,18495,18496,18497,18498,
18591,18592,18593,18594,18595,18596,18597,18598,18691,18692,18693,18694,18695,
18696,18697,18698,18791,18792,18793,18794,18795,18796,18797,18798,18891,18892,
18893,18894,18895,18896,18897,18898,18991,18992,18993,18994,18995,18996,18997,
18998,18911,18912,18913,18914,18915,18916,18917,18918,18921,18922,18923,18924,
18925,18926,18927,18928,18931,18932,18933,18934,18935,18936,18937,18938,18941,
18942,18943,18944,18945,18946,18947,18948,18951,18952,18953,18954,18955,18956,
18957,18958,18961,18962,18963,18964,18965,18966,18967,18968,18971,18972,18973,
18974,18975,18976,18977,18978,18981,18982,18983,18984,18985,18986,18987,18988,
18901,18902,18903,18904,18905,18906,18907,18908,19009,19019,19029,19039,19049,
19059,19069,19079,19089,19091,19092,19093,19094,19095,19096,19097,19098,19099,
19119,19129,19139,19149,19159,19169,19179,19189,19191,19192,19193,19194,19195,
19196,19197,19198,19199,19219,19229,19239,19249,19259,19269,19279,19289,19291,
19292,19293,19294,19295,19296,19297,19298,19299,19319,19329,19339,19349,19359,
19369,19379,19389,19391,19392,19393,19394,19395,19396,19397,19398,19399,19419,
19429,19439,19449,19459,19469,19479,19489,19491,19492,19493,19494,19495,19496,
19497,19498,19499,19519,19529,19539,19549,19559,19569,19579,19589,19591,19592,
19593,19594,19595,19596,19597,19598,19599,19619,19629,19639,19649,19659,19669,
19679,19689,19691,19692,19693,19694,19695,19696,19697,19698,19699,19719,19729,
19739,19749,19759,19769,19779,19789,19791,19792,19793,19794,19795,19796,19797,
19798,19799,19819,19829,19839,19849,19859,19869,19879,19889,19891,19892,19893,
19894,19895,19896,19897,19898,19899,19911,19912,19913,19914,19915,19916,19917,
19918,19919,19921,19922,19923,19924,19925,19926,19927,19928,19929,19931,19932,
19933,19934,19935,19936,19937,19938,19939,19941,19942,19943,19944,19945,19946,
19947,19948,19949,19951,19952,19953,19954,19955,19956,19957,19958,19959,19961,
19962,19963,19964,19965,19966,19967,19968,19969,19971,19972,19973,19974,19975,
19976,19977,19978,19979,19981,19982,19983,19984,19985,19986,19987,19988,19989,
19991,19992,19993,19994,19995,19996,19997,19998,19999,19109,19209,19309,19409,
19509,19609,19709,19809,19901,19902,19903,19904,19905,19906,19907,19908,19909,
19001,19002,19003,19004,19005,19006,19007,19008,19021,19022,19023,19024,19025,
19026,19027,19028,19031,19032,19033,19034,19035,19036,19037,19038,19041,19042,
19043,19044,19045,19046,19047,19048,19051,19052,19053,19054,19055,19056,19057,
19058,19061,19062,19063,19064,19065,19066,19067,19068,19071,19072,19073,19074,
19075,19076,19077,19078,19081,19082,19083,19084,19085,19086,19087,19088,19101,

157


19102,19103,19104,19105,19106,19107,19108,19121,19122,19123,19124,19125,19126,
19127,19128,19131,19132,19133,19134,19135,19136,19137,19138,19141,19142,19143,
19144,19145,19146,19147,19148,19151,19152,19153,19154,19155,19156,19157,19158,
19161,19162,19163,19164,19165,19166,19167,19168,19171,19172,19173,19174,19175,
19176,19177,19178,19181,19182,19183,19184,19185,19186,19187,19188,19201,19202,
19203,19204,19205,19206,19207,19208,19221,19222,19223,19224,19225,19226,19227,
19228,19231,19232,19233,19234,19235,19236,19237,19238,19241,19242,19243,19244,
19245,19246,19247,19248,19251,19252,19253,19254,19255,19256,19257,19258,19261,
19262,19263,19264,19265,19266,19267,19268,19271,19272,19273,19274,19275,19276,
19277,19278,19281,19282,19283,19284,19285,19286,19287,19288,19301,19302,19303,
19304,19305,19306,19307,19308,19321,19322,19323,19324,19325,19326,19327,19328,
19331,19332,19333,19334,19335,19336,19337,19338,19341,19342,19343,19344,19345,
19346,19347,19348,19351,19352,19353,19354,19355,19356,19357,19358,19361,19362,
19363,19364,19365,19366,19367,19368,19371,19372,19373,19374,19375,19376,19377,
19378,19381,19382,19383,19384,19385,19386,19387,19388,19401,19402,19403,19404,
19405,19406,19407,19408,19421,19422,19423,19424,19425,19426,19427,19428,19431,
19432,19433,19434,19435,19436,19437,19438,19441,19442,19443,19444,19445,19446,
19447,19448,19451,19452,19453,19454,19455,19456,19457,19458,19461,19462,19463,
19464,19465,19466,19467,19468,19471,19472,19473,19474,19475,19476,19477,19478,
19481,19482,19483,19484,19485,19486,19487,19488,19501,19502,19503,19504,19505,
19506,19507,19508,19521,19522,19523,19524,19525,19526,19527,19528,19531,19532,
19533,19534,19535,19536,19537,19538,19541,19542,19543,19544,19545,19546,19547,
19548,19551,19552,19553,19554,19555,19556,19557,19558,19561,19562,19563,19564,
19565,19566,19567,19568,19571,19572,19573,19574,19575,19576,19577,19578,19581,
19582,19583,19584,19585,19586,19587,19588,19601,19602,19603,19604,19605,19606,
19607,19608,19621,19622,19623,19624,19625,19626,19627,19628,19631,19632,19633,
19634,19635,19636,19637,19638,19641,19642,19643,19644,19645,19646,19647,19648,
19651,19652,19653,19654,19655,19656,19657,19658,19661,19662,19663,19664,19665,
19666,19667,19668,19671,19672,19673,19674,19675,19676,19677,19678,19681,19682,
19683,19684,19685,19686,19687,19688,19701,19702,19703,19704,19705,19706,19707,
19708,19721,19722,19723,19724,19725,19726,19727,19728,19731,19732,19733,19734,
19735,19736,19737,19738,19741,19742,19743,19744,19745,19746,19747,19748,19751,
19752,19753,19754,19755,19756,19757,19758,19761,19762,19763,19764,19765,19766,
19767,19768,19771,19772,19773,19774,19775,19776,19777,19778,19781,19782,19783,
19784,19785,19786,19787,19788,19801,19802,19803,19804,19805,19806,19807,19808,
19821,19822,19823,19824,19825,19826,19827,19828,19831,19832,19833,19834,19835,
19836,19837,19838,19841,19842,19843,19844,19845,19846,19847,19848,19851,19852,
19853,19854,19855,19856,19857,19858,19861,19862,19863,19864,19865,19866,19867,
19868,19871,19872,19873,19874,19875,19876,19877,19878,19881,19882,19883,19884,
19885,19886,19887,19888}

158


{11019,11119,11129,11139,11149,11159,11169,11179,11189,11199,11219,11229,11239
,11249,11259,11269,11279,11289,11299,11319,11329,11339,11349,11359,11369,11379,
11389,11399,11419,11429,11439,11449,11459,11469,11479,11489,11499,11519,11529,
11539,11549,11559,11569,11579,11589,11599,11619,11629,11639,11649,11659,11669,
11679,11689,11699,11719,11729,11739,11749,11759,11769,11779,11789,11799,11819,
11829,11839,11849,11859,11869,11879,11889,11899,11911,11912,11913,11914,11915,
11916,11917,11918,11919,11921,11922,11923,11924,11925,11926,11927,11928,11929,
11931,11932,11933,11934,11935,11936,11937,11938,11939,11941,11942,11943,11944,
11945,11946,11947,11948,11949,11951,11952,11953,11954,11955,11956,11957,11958,
11959,11961,11962,11963,11964,11965,11966,11967,11968,11969,11971,11972,11973,
11974,11975,11976,11977,11978,11979,11981,11982,11983,11984,11985,11986,11987,
11988,11989,11991,11992,11993,11994,11995,11996,11997,11998,11999,12019,12119,
12129,12139,12149,12159,12169,12179,12189,12191,12192,12193,12194,12195,12196,
12197,12198,12199,12219,12229,12239,12249,12259,12269,12279,12289,12291,12292,
12293,12294,12295,12296,12297,12298,12299,12319,12329,12339,12349,12359,12369,
12379,12389,12391,12392,12393,12394,12395,12396,12397,12398,12399,12419,12429,
12439,12449,12459,12469,12479,12489,12491,12492,12493,12494,12495,12496,12497,
12498,12499,12519,12529,12539,12549,12559,12569,12579,12589,12591,12592,12593,
12594,12595,12596,12597,12598,12599,12619,12629,12639,12649,12659,12669,12679,
12689,12691,12692,12693,12694,12695,12696,12697,12698,12699,12719,12729,12739,
12749,12759,12769,12779,12789,12791,12792,12793,12794,12795,12796,12797,12798,
12799,12819,12829,12839,12849,12859,12869,12879,12889,12891,12892,12893,12894,
12895,12896,12897,12898,12899,12911,12912,12913,12914,12915,12916,12917,12918,
12919,12921,12922,12923,12924,12925,12926,12927,12928,12929,12931,12932,12933,
12934,12935,12936,12937,12938,12939,12941,12942,12943,12944,12945,12946,12947,
12948,12949,12951,12952,12953,12954,12955,12956,12957,12958,12959,12961,12962,
12963,12964,12965,12966,12967,12968,12969,12971,12972,12973,12974,12975,12976,
12977,12978,12979,12981,12982,12983,12984,12985,12986,12987,12988,12989,12991,
12992,12993,12994,12995,12996,12997,12998,12999,13019,13119,13129,13139,13149,
13159,13169,13179,13189,13191,13192,13193,13194,13195,13196,13197,13198,13199,
13219,13229,13239,13249,13259,13269,13279,13289,13291,13292,13293,13294,13295,
13296,13297,13298,13299,13319,13329,13339,13349,13359,13369,13379,13389,13391,
13392,13393,13394,13395,13396,13397,13398,13399,13419,13429,13439,13449,13459,
13469,13479,13489,13491,13492,13493,13494,13495,13496,13497,13498,13499,13519,
13529,13539,13549,13559,13569,13579,13589,13591,13592,13593,13594,13595,13596,
13597,13598,13599,13619,13629,13639,13649,13659,13669,13679,13689,13691,13692,
13693,13694,13695,13696,13697,13698,13699,13719,13729,13739,13749,13759,13769,
13779,13789,13791,13792,13793,13794,13795,13796,13797,13798,13799,13819,13829,
13839,13849,13859,13869,13879,13889,13891,13892,13893,13894,13895,13896,13897,
13898,13899,13911,13912,13913,13914,13915,13916,13917,13918,13919,13921,13922,
13923,13924,13925,13926,13927,13928,13929,13931,13932,13933,13934,13935,13936,
13937,13938,13939,13941,13942,13943,13944,13945,13946,13947,13948,13949,13951,
13952,13953,13954,13955,13956,13957,13958,13959,13961,13962,13963,13964,13965,

159


13966,13967,13968,13969,13971,13972,13973,13974,13975,13976,13977,13978,13979,
13981,13982,13983,13984,13985,13986,13987,13988,13989,13991,13992,13993,13994,
13995,13996,13997,13998,13999,14019,14119,14129,14139,14149,14159,14169,14179,
14189,14191,14192,14193,14194,14195,14196,14197,14198,14199,14219,14229,14239,
14249,14259,14269,14279,14289,14291,14292,14293,14294,14295,14296,14297,14298,
14299,14319,14329,14339,14349,14359,14369,14379,14389,14391,14392,14393,14394,
14395,14396,14397,14398,14399,14419,14429,14439,14449,14459,14469,14479,14489,
14491,14492,14493,14494,14495,14496,14497,14498,14499,14519,14529,14539,14549,
14559,14569,14579,14589,14591,14592,14593,14594,14595,14596,14597,14598,14599,
14619,14629,14639,14649,14659,14669,14679,14689,14691,14692,14693,14694,14695,
14696,14697,14698,14699,14719,14729,14739,14749,14759,14769,14779,14789,14791,
14792,14793,14794,14795,14796,14797,14798,14799,14819,14829,14839,14849,14859,
14869,14879,14889,14891,14892,14893,14894,14895,14896,14897,14898,14899,14911,
14912,14913,14914,14915,14916,14917,14918,14919,14921,14922,14923,14924,14925,
14926,14927,14928,14929,14931,14932,14933,14934,14935,14936,14937,14938,14939,
14941,14942,14943,14944,14945,14946,14947,14948,14949,14951,14952,14953,14954,
14955,14956,14957,14958,14959,14961,14962,14963,14964,14965,14966,14967,14968,
14969,14971,14972,14973,14974,14975,14976,14977,14978,14979,14981,14982,14983,
14984,14985,14986,14987,14988,14989,14991,14992,14993,14994,14995,14996,14997,
14998,14999,15019,15119,15129,15139,15149,15159,15169,15179,15189,15191,15192,
15193,15194,15195,15196,15197,15198,15199,15219,15229,15239,15249,15259,15269,
15279,15289,15291,15292,15293,15294,15295,15296,15297,15298,15299,15319,15329,
15339,15349,15359,15369,15379,15389,15391,15392,15393,15394,15395,15396,15397,
15398,15399,15419,15429,15439,15449,15459,15469,15479,15489,15491,15492,15493,
15494,15495,15496,15497,15498,15499,15519,15529,15539,15549,15559,15569,15579,
15589,15591,15592,15593,15594,15595,15596,15597,15598,15599,15619,15629,15639,
15649,15659,15669,15679,15689,15691,15692,15693,15694,15695,15696,15697,15698,
15699,15719,15729,15739,15749,15759,15769,15779,15789,15791,15792,15793,15794,
15795,15796,15797,15798,15799,15819,15829,15839,15849,15859,15869,15879,15889,
15891,15892,15893,15894,15895,15896,15897,15898,15899,15911,15912,15913,15914,
15915,15916,15917,15918,15919,15921,15922,15923,15924,15925,15926,15927,15928,
15929,15931,15932,15933,15934,15935,15936,15937,15938,15939,15941,15942,15943,
15944,15945,15946,15947,15948,15949,15951,15952,15953,15954,15955,15956,15957,
15958,15959,15961,15962,15963,15964,15965,15966,15967,15968,15969,15971,15972,
15973,15974,15975,15976,15977,15978,15979,15981,15982,15983,15984,15985,15986,
15987,15988,15989,15991,15992,15993,15994,15995,15996,15997,15998,15999,16019,
16119,16129,16139,16149,16159,16169,16179,16189,16191,16192,16193,16194,16195,
16196,16197,16198,16199,16219,16229,16239,16249,16259,16269,16279,16289,16291,
16292,16293,16294,16295,16296,16297,16298,16299,16319,16329,16339,16349,16359,
16369,16379,16389,16391,16392,16393,16394,16395,16396,16397,16398,16399,16419,
16429,16439,16449,16459,16469,16479,16489,16491,16492,16493,16494,16495,16496,
16497,16498,16499,16519,16529,16539,16549,16559,16569,16579,16589,16591,16592,
16593,16594,16595,16596,16597,16598,16599,16619,16629,16639,16649,16659,16669,
16679,16689,16691,16692,16693,16694,16695,16696,16697,16698,16699,16719,16729,
16739,16749,16759,16769,16779,16789,16791,16792,16793,16794,16795,16796,16797,
16798,16799,16819,16829,16839,16849,16859,16869,16879,16889,16891,16892,16893,

160


16894,16895,16896,16897,16898,16899,16911,16912,16913,16914,16915,16916,16917,
16918,16919,16921,16922,16923,16924,16925,16926,16927,16928,16929,16931,16932,
16933,16934,16935,16936,16937,16938,16939,16941,16942,16943,16944,16945,16946,
16947,16948,16949,16951,16952,16953,16954,16955,16956,16957,16958,16959,16961,
16962,16963,16964,16965,16966,16967,16968,16969,16971,16972,16973,16974,16975,
16976,16977,16978,16979,16981,16982,16983,16984,16985,16986,16987,16988,16989,
16991,16992,16993,16994,16995,16996,16997,16998,16999,17019,17119,17129,17139,
17149,17159,17169,17179,17189,17191,17192,17193,17194,17195,17196,17197,17198,
17199,17219,17229,17239,17249,17259,17269,17279,17289,17291,17292,17293,17294,
17295,17296,17297,17298,17299,17319,17329,17339,17349,17359,17369,17379,17389,
17391,17392,17393,17394,17395,17396,17397,17398,17399,17419,17429,17439,17449,
17459,17469,17479,17489,17491,17492,17493,17494,17495,17496,17497,17498,17499,
17519,17529,17539,17549,17559,17569,17579,17589,17591,17592,17593,17594,17595,
17596,17597,17598,17599,17619,17629,17639,17649,17659,17669,17679,17689,17691,
17692,17693,17694,17695,17696,17697,17698,17699,17719,17729,17739,17749,17759,
17769,17779,17789,17791,17792,17793,17794,17795,17796,17797,17798,17799,17819,
17829,17839,17849,17859,17869,17879,17889,17891,17892,17893,17894,17895,17896,
17897,17898,17899,17911,17912,17913,17914,17915,17916,17917,17918,17919,17921,
17922,17923,17924,17925,17926,17927,17928,17929,17931,17932,17933,17934,17935,
17936,17937,17938,17939,17941,17942,17943,17944,17945,17946,17947,17948,17949,
17951,17952,17953,17954,17955,17956,17957,17958,17959,17961,17962,17963,17964,
17965,17966,17967,17968,17969,17971,17972,17973,17974,17975,17976,17977,17978,
17979,17981,17982,17983,17984,17985,17986,17987,17988,17989,17991,17992,17993,
17994,17995,17996,17997,17998,17999,18019,18119,18129,18139,18149,18159,18169,
18179,18189,18191,18192,18193,18194,18195,18196,18197,18198,18199,18219,18229,
18239,18249,18259,18269,18279,18289,18291,18292,18293,18294,18295,18296,18297,
18298,18299,18319,18329,18339,18349,18359,18369,18379,18389,18391,18392,18393,
18394,18395,18396,18397,18398,18399,18419,18429,18439,18449,18459,18469,18479,
18489,18491,18492,18493,18494,18495,18496,18497,18498,18499,18519,18529,18539,
18549,18559,18569,18579,18589,18591,18592,18593,18594,18595,18596,18597,18598,
18599,18619,18629,18639,18649,18659,18669,18679,18689,18691,18692,18693,18694,
18695,18696,18697,18698,18699,18719,18729,18739,18749,18759,18769,18779,18789,
18791,18792,18793,18794,18795,18796,18797,18798,18799,18819,18829,18839,18849,
18859,18869,18879,18889,18891,18892,18893,18894,18895,18896,18897,18898,18899,
18911,18912,18913,18914,18915,18916,18917,18918,18919,18921,18922,18923,18924,
18925,18926,18927,18928,18929,18931,18932,18933,18934,18935,18936,18937,18938,
18939,18941,18942,18943,18944,18945,18946,18947,18948,18949,18951,18952,18953,
18954,18955,18956,18957,18958,18959,18961,18962,18963,18964,18965,18966,18967,
18968,18969,18971,18972,18973,18974,18975,18976,18977,18978,18979,18981,18982,
18983,18984,18985,18986,18987,18988,18989,18991,18992,18993,18994,18995,18996,
18997,18998,18999,19019,19119,19121,19122,19123,19124,19125,19126,19127,19128,
19129,19131,19132,19133,19134,19135,19136,19137,19138,19139,19141,19142,19143,
19144,19145,19146,19147,19148,19149,19151,19152,19153,19154,19155,19156,19157,
19158,19159,19161,19162,19163,19164,19165,19166,19167,19168,19169,19171,19172,
19173,19174,19175,19176,19177,19178,19179,19181,19182,19183,19184,19185,19186,
19187,19188,19189,19191,19192,19193,19194,19195,19196,19197,19198,19199,19219,

161


19221,19222,19223,19224,19225,19226,19227,19228,19229,19231,19232,19233,19234,
19235,19236,19237,19238,19239,19241,19242,19243,19244,19245,19246,19247,19248,
19249,19251,19252,19253,19254,19255,19256,19257,19258,19259,19261,19262,19263,
19264,19265,19266,19267,19268,19269,19271,19272,19273,19274,19275,19276,19277,
19278,19279,19281,19282,19283,19284,19285,19286,19287,19288,19289,19291,19292,
19293,19294,19295,19296,19297,19298,19299,19319,19321,19322,19323,19324,19325,
19326,19327,19328,19329,19331,19332,19333,19334,19335,19336,19337,19338,19339,
19341,19342,19343,19344,19345,19346,19347,19348,19349,19351,19352,19353,19354,
19355,19356,19357,19358,19359,19361,19362,19363,19364,19365,19366,19367,19368,
19369,19371,19372,19373,19374,19375,19376,19377,19378,19379,19381,19382,19383,
19384,19385,19386,19387,19388,19389,19391,19392,19393,19394,19395,19396,19397,
19398,19399,19419,19421,19422,19423,19424,19425,19426,19427,19428,19429,19431,
19432,19433,19434,19435,19436,19437,19438,19439,19441,19442,19443,19444,19445,
19446,19447,19448,19449,19451,19452,19453,19454,19455,19456,19457,19458,19459,
19461,19462,19463,19464,19465,19466,19467,19468,19469,19471,19472,19473,19474,
19475,19476,19477,19478,19479,19481,19482,19483,19484,19485,19486,19487,19488,
19489,19491,19492,19493,19494,19495,19496,19497,19498,19499,19519,19521,19522,
19523,19524,19525,19526,19527,19528,19529,19531,19532,19533,19534,19535,19536,
19537,19538,19539,19541,19542,19543,19544,19545,19546,19547,19548,19549,19551,
19552,19553,19554,19555,19556,19557,19558,19559,19561,19562,19563,19564,19565,
19566,19567,19568,19569,19571,19572,19573,19574,19575,19576,19577,19578,19579,
19581,19582,19583,19584,19585,19586,19587,19588,19589,19591,19592,19593,19594,
19595,19596,19597,19598,19599,19619,19621,19622,19623,19624,19625,19626,19627,
19628,19629,19631,19632,19633,19634,19635,19636,19637,19638,19639,19641,19642,
19643,19644,19645,19646,19647,19648,19649,19651,19652,19653,19654,19655,19656,
19657,19658,19659,19661,19662,19663,19664,19665,19666,19667,19668,19669,19671,
19672,19673,19674,19675,19676,19677,19678,19679,19681,19682,19683,19684,19685,
19686,19687,19688,19689,19691,19692,19693,19694,19695,19696,19697,19698,19699,
19719,19721,19722,19723,19724,19725,19726,19727,19728,19729,19731,19732,19733,
19734,19735,19736,19737,19738,19739,19741,19742,19743,19744,19745,19746,19747,
19748,19749,19751,19752,19753,19754,19755,19756,19757,19758,19759,19761,19762,
19763,19764,19765,19766,19767,19768,19769,19771,19772,19773,19774,19775,19776,
19777,19778,19779,19781,19782,19783,19784,19785,19786,19787,19788,19789,19791,
19792,19793,19794,19795,19796,19797,19798,19799,19819,19821,19822,19823,19824,
19825,19826,19827,19828,19829,19831,19832,19833,19834,19835,19836,19837,19838,
19839,19841,19842,19843,19844,19845,19846,19847,19848,19849,19851,19852,19853,
19854,19855,19856,19857,19858,19859,19861,19862,19863,19864,19865,19866,19867,
19868,19869,19871,19872,19873,19874,19875,19876,19877,19878,19879,19881,19882,
19883,19884,19885,19886,19887,19888,19889,19891,19892,19893,19894,19895,19896,
19897,19898,19899,19911,19912,19913,19914,19915,19916,19917,19918,19919,19921,
19922,19923,19924,19925,19926,19927,19928,19929,19931,19932,19933,19934,19935,
19936,19937,19938,19939,19941,19942,19943,19944,19945,19946,19947,19948,19949,
19951,19952,19953,19954,19955,19956,19957,19958,19959,19961,19962,19963,19964,
19965,19966,19967,19968,19969,19971,19972,19973,19974,19975,19976,19977,19978,
19979,19981,19982,19983,19984,19985,19986,19987,19988,19989,19991,19992,19993,
19994,19995,19996,19997,19998,19999}

162


Suma dismal entre primos primarios de 5 cifras con secundarios de 4 cifras

{11009,11019,11029,11039,11049,11059,11069,11079,11089,11099,11119,11129,11139
,11149,11159,11169,11179,11189,11199,11219,11229,11239,11249,11259,11269,11279,
11289,11299,11319,11329,11339,11349,11359,11369,11379,11389,11399,11419,11429,
11439,11449,11459,11469,11479,11489,11499,11519,11529,11539,11549,11559,11569,
11579,11589,11599,11619,11629,11639,11649,11659,11669,11679,11689,11699,11719,
11729,11739,11749,11759,11769,11779,11789,11799,11819,11829,11839,11849,11859,
11869,11879,11889,11899,11919,11929,11939,11949,11959,11969,11979,11989,11999,
11109,11209,11309,11409,11509,11609,11709,11809,11909,12009,12029,12039,12049,
12059,12069,12079,12089,12099,12109,12209,12309,12409,12509,12609,12709,12809,
12909,13009,13029,13039,13049,13059,13069,13079,13089,13099,13109,13209,13309,
13409,13509,13609,13709,13809,13909,14009,14029,14039,14049,14059,14069,14079,
14089,14099,14109,14209,14309,14409,14509,14609,14709,14809,14909,15009,15029,
15039,15049,15059,15069,15079,15089,15099,15109,15209,15309,15409,15509,15609,
15709,15809,15909,16009,16029,16039,16049,16059,16069,16079,16089,16099,16109,
16209,16309,16409,16509,16609,16709,16809,16909,17009,17029,17039,17049,17059,
17069,17079,17089,17099,17109,17209,17309,17409,17509,17609,17709,17809,17909,
18009,18029,18039,18049,18059,18069,18079,18089,18099,18109,18209,18309,18409,
18509,18609,18709,18809,18909,19009,19029,19039,19049,19059,19069,19079,19089,
19099,19109,19209,19309,19409,19509,19609,19709,19809,19909,12019,12119,12219,
12319,12419,12519,12619,12719,12819,12919,13019,13119,13219,13319,13419,13519,
13619,13719,13819,13919,14019,14119,14219,14319,14419,14519,14619,14719,14819,
14919,15019,15119,15219,15319,15419,15519,15619,15719,15819,15919,16019,16119,
16219,16319,16419,16519,16619,16719,16819,16919,17019,17119,17219,17319,17419,
17519,17619,17719,17819,17919,18019,18119,18219,18319,18419,18519,18619,18719,
18819,18919,19019,19119,19219,19319,19419,19519,19619,19719,19819,19919,12129,
12229,12329,12429,12529,12629,12729,12829,12929,13129,13229,13329,13429,13529,
13629,13729,13829,13929,14129,14229,14329,14429,14529,14629,14729,14829,14929,
15129,15229,15329,15429,15529,15629,15729,15829,15929,16129,16229,16329,16429,
16529,16629,16729,16829,16929,17129,17229,17329,17429,17529,17629,17729,17829,
17929,18129,18229,18329,18429,18529,18629,18729,18829,18929,19129,19229,19329,
19429,19529,19629,19729,19829,19929,12139,12239,12339,12439,12539,12639,12739,
12839,12939,13139,13239,13339,13439,13539,13639,13739,13839,13939,14139,14239,
14339,14439,14539,14639,14739,14839,14939,15139,15239,15339,15439,15539,15639,
15739,15839,15939,16139,16239,16339,16439,16539,16639,16739,16839,16939,17139,
17239,17339,17439,17539,17639,17739,17839,17939,18139,18239,18339,18439,18539,

163


18639,18739,18839,18939,19139,19239,19339,19439,19539,19639,19739,19839,19939,
12149,12249,12349,12449,12549,12649,12749,12849,12949,13149,13249,13349,13449,
13549,13649,13749,13849,13949,14149,14249,14349,14449,14549,14649,14749,14849,
14949,15149,15249,15349,15449,15549,15649,15749,15849,15949,16149,16249,16349,
16449,16549,16649,16749,16849,16949,17149,17249,17349,17449,17549,17649,17749,
17849,17949,18149,18249,18349,18449,18549,18649,18749,18849,18949,19149,19249,
19349,19449,19549,19649,19749,19849,19949,12159,12259,12359,12459,12559,12659,
12759,12859,12959,13159,13259,13359,13459,13559,13659,13759,13859,13959,14159,
14259,14359,14459,14559,14659,14759,14859,14959,15159,15259,15359,15459,15559,
15659,15759,15859,15959,16159,16259,16359,16459,16559,16659,16759,16859,16959,
17159,17259,17359,17459,17559,17659,17759,17859,17959,18159,18259,18359,18459,
18559,18659,18759,18859,18959,19159,19259,19359,19459,19559,19659,19759,19859,
19959,12169,12269,12369,12469,12569,12669,12769,12869,12969,13169,13269,13369,
13469,13569,13669,13769,13869,13969,14169,14269,14369,14469,14569,14669,14769,
14869,14969,15169,15269,15369,15469,15569,15669,15769,15869,15969,16169,16269,
16369,16469,16569,16669,16769,16869,16969,17169,17269,17369,17469,17569,17669,
17769,17869,17969,18169,18269,18369,18469,18569,18669,18769,18869,18969,19169,
19269,19369,19469,19569,19669,19769,19869,19969,12179,12279,12379,12479,12579,
12679,12779,12879,12979,13179,13279,13379,13479,13579,13679,13779,13879,13979,
14179,14279,14379,14479,14579,14679,14779,14879,14979,15179,15279,15379,15479,
15579,15679,15779,15879,15979,16179,16279,16379,16479,16579,16679,16779,16879,
16979,17179,17279,17379,17479,17579,17679,17779,17879,17979,18179,18279,18379,
18479,18579,18679,18779,18879,18979,19179,19279,19379,19479,19579,19679,19779,
19879,19979,12189,12289,12389,12489,12589,12689,12789,12889,12989,13189,13289,
13389,13489,13589,13689,13789,13889,13989,14189,14289,14389,14489,14589,14689,
14789,14889,14989,15189,15289,15389,15489,15589,15689,15789,15889,15989,16189,
16289,16389,16489,16589,16689,16789,16889,16989,17189,17289,17389,17489,17589,
17689,17789,17889,17989,18189,18289,18389,18489,18589,18689,18789,18889,18989,
19189,19289,19389,19489,19589,19689,19789,19889,19989,11091,11092,11093,11094,
11095,11096,11097,11098,11191,11192,11193,11194,11195,11196,11197,11198,11291,
11292,11293,11294,11295,11296,11297,11298,11391,11392,11393,11394,11395,11396,
11397,11398,11491,11492,11493,11494,11495,11496,11497,11498,11591,11592,11593,
11594,11595,11596,11597,11598,11691,11692,11693,11694,11695,11696,11697,11698,
11791,11792,11793,11794,11795,11796,11797,11798,11891,11892,11893,11894,11895,
11896,11897,11898,11991,11992,11993,11994,11995,11996,11997,11998,12091,12092,
12093,12094,12095,12096,12097,12098,12199,12299,12399,12499,12599,12699,12799,
12899,12991,12992,12993,12994,12995,12996,12997,12998,12999,13091,13092,13093,
13094,13095,13096,13097,13098,13199,13299,13399,13499,13599,13699,13799,13899,
13991,13992,13993,13994,13995,13996,13997,13998,13999,14091,14092,14093,14094,
14095,14096,14097,14098,14199,14299,14399,14499,14599,14699,14799,14899,14991,
14992,14993,14994,14995,14996,14997,14998,14999,15091,15092,15093,15094,15095,
15096,15097,15098,15199,15299,15399,15499,15599,15699,15799,15899,15991,15992,
15993,15994,15995,15996,15997,15998,15999,16091,16092,16093,16094,16095,16096,
16097,16098,16199,16299,16399,16499,16599,16699,16799,16899,16991,16992,16993,
16994,16995,16996,16997,16998,16999,17091,17092,17093,17094,17095,17096,17097,
17098,17199,17299,17399,17499,17599,17699,17799,17899,17991,17992,17993,17994,

164


17995,17996,17997,17998,17999,18091,18092,18093,18094,18095,18096,18097,18098,
18199,18299,18399,18499,18599,18699,18799,18899,18991,18992,18993,18994,18995,
18996,18997,18998,18999,19091,19092,19093,19094,19095,19096,19097,19098,19191,
19192,19193,19194,19195,19196,19197,19198,19199,19291,19292,19293,19294,19295,
19296,19297,19298,19299,19391,19392,19393,19394,19395,19396,19397,19398,19399,
19491,19492,19493,19494,19495,19496,19497,19498,19499,19591,19592,19593,19594,
19595,19596,19597,19598,19599,19691,19692,19693,19694,19695,19696,19697,19698,
19699,19791,19792,19793,19794,19795,19796,19797,19798,19799,19891,19892,19893,
19894,19895,19896,19897,19898,19899,19991,19992,19993,19994,19995,19996,19997,
19998,19999,11911,11912,11913,11914,11915,11916,11917,11918,11921,11922,11923,
11924,11925,11926,11927,11928,11931,11932,11933,11934,11935,11936,11937,11938,
11941,11942,11943,11944,11945,11946,11947,11948,11951,11952,11953,11954,11955,
11956,11957,11958,11961,11962,11963,11964,11965,11966,11967,11968,11971,11972,
11973,11974,11975,11976,11977,11978,11981,11982,11983,11984,11985,11986,11987,
11988,11901,11902,11903,11904,11905,11906,11907,11908,12901,12902,12903,12904,
12905,12906,12907,12908,13901,13902,13903,13904,13905,13906,13907,13908,14901,
14902,14903,14904,14905,14906,14907,14908,15901,15902,15903,15904,15905,15906,
15907,15908,16901,16902,16903,16904,16905,16906,16907,16908,17901,17902,17903,
17904,17905,17906,17907,17908,18901,18902,18903,18904,18905,18906,18907,18908,
19901,19902,19903,19904,19905,19906,19907,19908,19921,19922,19923,19924,19925,
19926,19927,19928,19931,19932,19933,19934,19935,19936,19937,19938,19941,19942,
19943,19944,19945,19946,19947,19948,19951,19952,19953,19954,19955,19956,19957,
19958,19961,19962,19963,19964,19965,19966,19967,19968,19971,19972,19973,19974,
19975,19976,19977,19978,19981,19982,19983,19984,19985,19986,19987,19988}

{11019,11119,11129,11139,11149,11159,11169,11179,11189,11191,11192,11193,11194
,11195,11196,11197,11198,11199,11219,11229,11239,11249,11259,11269,11279,11289,
11291,11292,11293,11294,11295,11296,11297,11298,11299,11319,11329,11339,11349,
11359,11369,11379,11389,11391,11392,11393,11394,11395,11396,11397,11398,11399,
11419,11429,11439,11449,11459,11469,11479,11489,11491,11492,11493,11494,11495,
11496,11497,11498,11499,11519,11529,11539,11549,11559,11569,11579,11589,11591,
11592,11593,11594,11595,11596,11597,11598,11599,11619,11629,11639,11649,11659,
11669,11679,11689,11691,11692,11693,11694,11695,11696,11697,11698,11699,11719,
11729,11739,11749,11759,11769,11779,11789,11791,11792,11793,11794,11795,11796,
11797,11798,11799,11819,11829,11839,11849,11859,11869,11879,11889,11891,11892,
11893,11894,11895,11896,11897,11898,11899,11911,11912,11913,11914,11915,11916,
11917,11918,11919,11921,11922,11923,11924,11925,11926,11927,11928,11929,11931,
11932,11933,11934,11935,11936,11937,11938,11939,11941,11942,11943,11944,11945,
11946,11947,11948,11949,11951,11952,11953,11954,11955,11956,11957,11958,11959,
11961,11962,11963,11964,11965,11966,11967,11968,11969,11971,11972,11973,11974,

165


11975,11976,11977,11978,11979,11981,11982,11983,11984,11985,11986,11987,11988,
11989,11991,11992,11993,11994,11995,11996,11997,11998,11999,12019,12119,12129,
12139,12149,12159,12169,12179,12189,12199,12219,12229,12239,12249,12259,12269,
12279,12289,12299,12319,12329,12339,12349,12359,12369,12379,12389,12399,12419,
12429,12439,12449,12459,12469,12479,12489,12499,12519,12529,12539,12549,12559,
12569,12579,12589,12599,12619,12629,12639,12649,12659,12669,12679,12689,12699,
12719,12729,12739,12749,12759,12769,12779,12789,12799,12819,12829,12839,12849,
12859,12869,12879,12889,12899,12919,12929,12939,12949,12959,12969,12979,12989,
12991,12992,12993,12994,12995,12996,12997,12998,12999,13019,13119,13129,13139,
13149,13159,13169,13179,13189,13199,13219,13229,13239,13249,13259,13269,13279,
13289,13299,13319,13329,13339,13349,13359,13369,13379,13389,13399,13419,13429,
13439,13449,13459,13469,13479,13489,13499,13519,13529,13539,13549,13559,13569,
13579,13589,13599,13619,13629,13639,13649,13659,13669,13679,13689,13699,13719,
13729,13739,13749,13759,13769,13779,13789,13799,13819,13829,13839,13849,13859,
13869,13879,13889,13899,13919,13929,13939,13949,13959,13969,13979,13989,13991,
13992,13993,13994,13995,13996,13997,13998,13999,14019,14119,14129,14139,14149,
14159,14169,14179,14189,14199,14219,14229,14239,14249,14259,14269,14279,14289,
14299,14319,14329,14339,14349,14359,14369,14379,14389,14399,14419,14429,14439,
14449,14459,14469,14479,14489,14499,14519,14529,14539,14549,14559,14569,14579,
14589,14599,14619,14629,14639,14649,14659,14669,14679,14689,14699,14719,14729,
14739,14749,14759,14769,14779,14789,14799,14819,14829,14839,14849,14859,14869,
14879,14889,14899,14919,14929,14939,14949,14959,14969,14979,14989,14991,14992,
14993,14994,14995,14996,14997,14998,14999,15019,15119,15129,15139,15149,15159,
15169,15179,15189,15199,15219,15229,15239,15249,15259,15269,15279,15289,15299,
15319,15329,15339,15349,15359,15369,15379,15389,15399,15419,15429,15439,15449,
15459,15469,15479,15489,15499,15519,15529,15539,15549,15559,15569,15579,15589,
15599,15619,15629,15639,15649,15659,15669,15679,15689,15699,15719,15729,15739,
15749,15759,15769,15779,15789,15799,15819,15829,15839,15849,15859,15869,15879,
15889,15899,15919,15929,15939,15949,15959,15969,15979,15989,15991,15992,15993,
15994,15995,15996,15997,15998,15999,16019,16119,16129,16139,16149,16159,16169,
16179,16189,16199,16219,16229,16239,16249,16259,16269,16279,16289,16299,16319,
16329,16339,16349,16359,16369,16379,16389,16399,16419,16429,16439,16449,16459,
16469,16479,16489,16499,16519,16529,16539,16549,16559,16569,16579,16589,16599,
16619,16629,16639,16649,16659,16669,16679,16689,16699,16719,16729,16739,16749,
16759,16769,16779,16789,16799,16819,16829,16839,16849,16859,16869,16879,16889,
16899,16919,16929,16939,16949,16959,16969,16979,16989,16991,16992,16993,16994,
16995,16996,16997,16998,16999,17019,17119,17129,17139,17149,17159,17169,17179,
17189,17199,17219,17229,17239,17249,17259,17269,17279,17289,17299,17319,17329,
17339,17349,17359,17369,17379,17389,17399,17419,17429,17439,17449,17459,17469,
17479,17489,17499,17519,17529,17539,17549,17559,17569,17579,17589,17599,17619,
17629,17639,17649,17659,17669,17679,17689,17699,17719,17729,17739,17749,17759,
17769,17779,17789,17799,17819,17829,17839,17849,17859,17869,17879,17889,17899,
17919,17929,17939,17949,17959,17969,17979,17989,17991,17992,17993,17994,17995,
17996,17997,17998,17999,18019,18119,18129,18139,18149,18159,18169,18179,18189,
18199,18219,18229,18239,18249,18259,18269,18279,18289,18299,18319,18329,18339,
18349,18359,18369,18379,18389,18399,18419,18429,18439,18449,18459,18469,18479,

166


18489,18499,18519,18529,18539,18549,18559,18569,18579,18589,18599,18619,18629,
18639,18649,18659,18669,18679,18689,18699,18719,18729,18739,18749,18759,18769,
18779,18789,18799,18819,18829,18839,18849,18859,18869,18879,18889,18899,18919,
18929,18939,18949,18959,18969,18979,18989,18991,18992,18993,18994,18995,18996,
18997,18998,18999,19019,19119,19129,19139,19149,19159,19169,19179,19189,19191,
19192,19193,19194,19195,19196,19197,19198,19199,19219,19229,19239,19249,19259,
19269,19279,19289,19291,19292,19293,19294,19295,19296,19297,19298,19299,19319,
19329,19339,19349,19359,19369,19379,19389,19391,19392,19393,19394,19395,19396,
19397,19398,19399,19419,19429,19439,19449,19459,19469,19479,19489,19491,19492,
19493,19494,19495,19496,19497,19498,19499,19519,19529,19539,19549,19559,19569,
19579,19589,19591,19592,19593,19594,19595,19596,19597,19598,19599,19619,19629,
19639,19649,19659,19669,19679,19689,19691,19692,19693,19694,19695,19696,19697,
19698,19699,19719,19729,19739,19749,19759,19769,19779,19789,19791,19792,19793,
19794,19795,19796,19797,19798,19799,19819,19829,19839,19849,19859,19869,19879,
19889,19891,19892,19893,19894,19895,19896,19897,19898,19899,19919,19921,19922,
19923,19924,19925,19926,19927,19928,19929,19931,19932,19933,19934,19935,19936,
19937,19938,19939,19941,19942,19943,19944,19945,19946,19947,19948,19949,19951,
19952,19953,19954,19955,19956,19957,19958,19959,19961,19962,19963,19964,19965,
19966,19967,19968,19969,19971,19972,19973,19974,19975,19976,19977,19978,19979,
19981,19982,19983,19984,19985,19986,19987,19988,19989,19991,19992,19993,19994,
19995,19996,19997,19998,19999}


{10109,10119,10129,10139,10149,10159,10169,10179,10189,10199,10219,10229,10239
,10249,10259,10269,10279,10289,10299,10319,10329,10339,10349,10359,10369,10379,
10389,10399,10419,10429,10439,10449,10459,10469,10479,10489,10499,10519,10529,
10539,10549,10559,10569,10579,10589,10599,10619,10629,10639,10649,10659,10669,
10679,10689,10699,10719,10729,10739,10749,10759,10769,10779,10789,10799,10819,
10829,10839,10849,10859,10869,10879,10889,10899,10919,10929,10939,10949,10959,
10969,10979,10989,10999,11109,11209,11309,11409,11509,11609,11709,11809,11909,
12109,12129,12139,12149,12159,12169,12179,12189,12199,12209,12309,12409,12509,
12609,12709,12809,12909,13109,13129,13139,13149,13159,13169,13179,13189,13199,
13209,13309,13409,13509,13609,13709,13809,13909,14109,14129,14139,14149,14159,
14169,14179,14189,14199,14209,14309,14409,14509,14609,14709,14809,14909,15109,
15129,15139,15149,15159,15169,15179,15189,15199,15209,15309,15409,15509,15609,
15709,15809,15909,16109,16129,16139,16149,16159,16169,16179,16189,16199,16209,

167


16309,16409,16509,16609,16709,16809,16909,17109,17129,17139,17149,17159,17169,
17179,17189,17199,17209,17309,17409,17509,17609,17709,17809,17909,18109,18129,
18139,18149,18159,18169,18179,18189,18199,18209,18309,18409,18509,18609,18709,
18809,18909,19109,19129,19139,19149,19159,19169,19179,19189,19199,19209,19309,
19409,19509,19609,19709,19809,19909,10209,12229,12239,12249,12259,12269,12279,
12289,12299,13229,13239,13249,13259,13269,13279,13289,13299,14229,14239,14249,
14259,14269,14279,14289,14299,15229,15239,15249,15259,15269,15279,15289,15299,
16229,16239,16249,16259,16269,16279,16289,16299,17229,17239,17249,17259,17269,
17279,17289,17299,18229,18239,18249,18259,18269,18279,18289,18299,19229,19239,
19249,19259,19269,19279,19289,19299,11219,11319,11419,11519,11619,11719,11819,
11919,12219,12319,12419,12519,12619,12719,12819,12919,13219,13319,13419,13519,
13619,13719,13819,13919,14219,14319,14419,14519,14619,14719,14819,14919,15219,
15319,15419,15519,15619,15719,15819,15919,16219,16319,16419,16519,16619,16719,
16819,16919,17219,17319,17419,17519,17619,17719,17819,17919,18219,18319,18419,
18519,18619,18719,18819,18919,19219,19319,19419,19519,19619,19719,19819,19919,
10309,12329,12339,12349,12359,12369,12379,12389,12399,13329,13339,13349,13359,
13369,13379,13389,13399,14329,14339,14349,14359,14369,14379,14389,14399,15329,
15339,15349,15359,15369,15379,15389,15399,16329,16339,16349,16359,16369,16379,
16389,16399,17329,17339,17349,17359,17369,17379,17389,17399,18329,18339,18349,
18359,18369,18379,18389,18399,19329,19339,19349,19359,19369,19379,19389,19399,
11329,11429,11529,11629,11729,11829,11929,12429,12529,12629,12729,12829,12929,
13429,13529,13629,13729,13829,13929,14429,14529,14629,14729,14829,14929,15429,
15529,15629,15729,15829,15929,16429,16529,16629,16729,16829,16929,17429,17529,
17629,17729,17829,17929,18429,18529,18629,18729,18829,18929,19429,19529,19629,
19729,19829,19929,10409,12439,12449,12459,12469,12479,12489,12499,13439,13449,
13459,13469,13479,13489,13499,14439,14449,14459,14469,14479,14489,14499,15439,
15449,15459,15469,15479,15489,15499,16439,16449,16459,16469,16479,16489,16499,
17439,17449,17459,17469,17479,17489,17499,18439,18449,18459,18469,18479,18489,
18499,19439,19449,19459,19469,19479,19489,19499,11439,11539,11639,11739,11839,
11939,12539,12639,12739,12839,12939,13539,13639,13739,13839,13939,14539,14639,
14739,14839,14939,15539,15639,15739,15839,15939,16539,16639,16739,16839,16939,
17539,17639,17739,17839,17939,18539,18639,18739,18839,18939,19539,19639,19739,
19839,19939,10509,12549,12559,12569,12579,12589,12599,13549,13559,13569,13579,
13589,13599,14549,14559,14569,14579,14589,14599,15549,15559,15569,15579,15589,
15599,16549,16559,16569,16579,16589,16599,17549,17559,17569,17579,17589,17599,
18549,18559,18569,18579,18589,18599,19549,19559,19569,19579,19589,19599,11549,
11649,11749,11849,11949,12649,12749,12849,12949,13649,13749,13849,13949,14649,
14749,14849,14949,15649,15749,15849,15949,16649,16749,16849,16949,17649,17749,
17849,17949,18649,18749,18849,18949,19649,19749,19849,19949,10609,12659,12669,
12679,12689,12699,13659,13669,13679,13689,13699,14659,14669,14679,14689,14699,
15659,15669,15679,15689,15699,16659,16669,16679,16689,16699,17659,17669,17679,
17689,17699,18659,18669,18679,18689,18699,19659,19669,19679,19689,19699,11659,
11759,11859,11959,12759,12859,12959,13759,13859,13959,14759,14859,14959,15759,
15859,15959,16759,16859,16959,17759,17859,17959,18759,18859,18959,19759,19859,
19959,10709,12769,12779,12789,12799,13769,13779,13789,13799,14769,14779,14789,
14799,15769,15779,15789,15799,16769,16779,16789,16799,17769,17779,17789,17799,

168


18769,18779,18789,18799,19769,19779,19789,19799,11769,11869,11969,12869,12969,
13869,13969,14869,14969,15869,15969,16869,16969,17869,17969,18869,18969,19869,
19969,10809,12879,12889,12899,13879,13889,13899,14879,14889,14899,15879,15889,
15899,16879,16889,16899,17879,17889,17899,18879,18889,18899,19879,19889,19899,
11879,11979,12979,13979,14979,15979,16979,17979,18979,19979,10909,10991,10992,
10993,10994,10995,10996,10997,10998,10911,10912,10913,10914,10915,10916,10917,
10918,10921,10922,10923,10924,10925,10926,10927,10928,10931,10932,10933,10934,
10935,10936,10937,10938,10941,10942,10943,10944,10945,10946,10947,10948,10951,
10952,10953,10954,10955,10956,10957,10958,10961,10962,10963,10964,10965,10966,
10967,10968,10971,10972,10973,10974,10975,10976,10977,10978,10981,10982,10983,
10984,10985,10986,10987,10988,11901,11902,11903,11904,11905,11906,11907,11908,
12989,12991,12992,12993,12994,12995,12996,12997,12998,12999,12901,12902,12903,
12904,12905,12906,12907,12908,13989,13991,13992,13993,13994,13995,13996,13997,
13998,13999,13901,13902,13903,13904,13905,13906,13907,13908,14989,14991,14992,
14993,14994,14995,14996,14997,14998,14999,14901,14902,14903,14904,14905,14906,
14907,14908,15989,15991,15992,15993,15994,15995,15996,15997,15998,15999,15901,
15902,15903,15904,15905,15906,15907,15908,16989,16991,16992,16993,16994,16995,
16996,16997,16998,16999,16901,16902,16903,16904,16905,16906,16907,16908,17989,
17991,17992,17993,17994,17995,17996,17997,17998,17999,17901,17902,17903,17904,
17905,17906,17907,17908,18989,18991,18992,18993,18994,18995,18996,18997,18998,
18999,18901,18902,18903,18904,18905,18906,18907,18908,19901,19902,19903,19904,
19905,19906,19907,19908,19921,19922,19923,19924,19925,19926,19927,19928,19931,
19932,19933,19934,19935,19936,19937,19938,19941,19942,19943,19944,19945,19946,
19947,19948,19951,19952,19953,19954,19955,19956,19957,19958,19961,19962,19963,
19964,19965,19966,19967,19968,19971,19972,19973,19974,19975,19976,19977,19978,
19981,19982,19983,19984,19985,19986,19987,19988,19989,19991,19992,19993,19994,
19995,19996,19997,19998,19999,11912,11913,11914,11915,11916,11917,11918,12912,
12913,12914,12915,12916,12917,12918,13912,13913,13914,13915,13916,13917,13918,
14912,14913,14914,14915,14916,14917,14918,15912,15913,15914,15915,15916,15917,
15918,16912,16913,16914,16915,16916,16917,16918,17912,17913,17914,17915,17916,
17917,17918,18912,18913,18914,18915,18916,18917,18918,19912,19913,19914,19915,
19916,19917,19918,11923,11924,11925,11926,11927,11928,12923,12924,12925,12926,
12927,12928,13923,13924,13925,13926,13927,13928,14923,14924,14925,14926,14927,
14928,15923,15924,15925,15926,15927,15928,16923,16924,16925,16926,16927,16928,
17923,17924,17925,17926,17927,17928,18923,18924,18925,18926,18927,18928,11934,
11935,11936,11937,11938,12934,12935,12936,12937,12938,13934,13935,13936,13937,
13938,14934,14935,14936,14937,14938,15934,15935,15936,15937,15938,16934,16935,
16936,16937,16938,17934,17935,17936,17937,17938,18934,18935,18936,18937,18938,
11945,11946,11947,11948,12945,12946,12947,12948,13945,13946,13947,13948,14945,
14946,14947,14948,15945,15946,15947,15948,16945,16946,16947,16948,17945,17946,
17947,17948,18945,18946,18947,18948,11956,11957,11958,12956,12957,12958,13956,
13957,13958,14956,14957,14958,15956,15957,15958,16956,16957,16958,17956,17957,
17958,18956,18957,18958,11967,11968,12967,12968,13967,13968,14967,14968,15967,
15968,16967,16968,17967,17968,18967,18968,11978,12978,13978,14978,15978,16978,
17978,18978,11989}

169


{11219,11319,11329,11419,11429,11439,11519,11529,11539,11549,11619,11629,11639
,11649,11659,11719,11729,11739,11749,11759,11769,11819,11829,11839,11849,11859,
11869,11879,11912,11913,11914,11915,11916,11917,11918,11919,11923,11924,11925,
11926,11927,11928,11929,11934,11935,11936,11937,11938,11939,11945,11946,11947,
11948,11949,11956,11957,11958,11959,11967,11968,11969,11978,11979,11989,12129,
12139,12149,12159,12169,12179,12189,12199,12219,12229,12239,12249,12259,12269,
12279,12289,12299,12319,12329,12339,12349,12359,12369,12379,12389,12399,12419,
12429,12439,12449,12459,12469,12479,12489,12499,12519,12529,12539,12549,12559,
12569,12579,12589,12599,12619,12629,12639,12649,12659,12669,12679,12689,12699,
12719,12729,12739,12749,12759,12769,12779,12789,12799,12819,12829,12839,12849,
12859,12869,12879,12889,12899,12912,12913,12914,12915,12916,12917,12918,12919,
12923,12924,12925,12926,12927,12928,12929,12934,12935,12936,12937,12938,12939,
12945,12946,12947,12948,12949,12956,12957,12958,12959,12967,12968,12969,12978,
12979,12989,12991,12992,12993,12994,12995,12996,12997,12998,12999,13129,13139,
13149,13159,13169,13179,13189,13199,13219,13229,13239,13249,13259,13269,13279,
13289,13299,13319,13329,13339,13349,13359,13369,13379,13389,13399,13419,13429,
13439,13449,13459,13469,13479,13489,13499,13519,13529,13539,13549,13559,13569,
13579,13589,13599,13619,13629,13639,13649,13659,13669,13679,13689,13699,13719,
13729,13739,13749,13759,13769,13779,13789,13799,13819,13829,13839,13849,13859,
13869,13879,13889,13899,13912,13913,13914,13915,13916,13917,13918,13919,13923,
13924,13925,13926,13927,13928,13929,13934,13935,13936,13937,13938,13939,13945,
13946,13947,13948,13949,13956,13957,13958,13959,13967,13968,13969,13978,13979,
13989,13991,13992,13993,13994,13995,13996,13997,13998,13999,14129,14139,14149,
14159,14169,14179,14189,14199,14219,14229,14239,14249,14259,14269,14279,14289,
14299,14319,14329,14339,14349,14359,14369,14379,14389,14399,14419,14429,14439,
14449,14459,14469,14479,14489,14499,14519,14529,14539,14549,14559,14569,14579,
14589,14599,14619,14629,14639,14649,14659,14669,14679,14689,14699,14719,14729,
14739,14749,14759,14769,14779,14789,14799,14819,14829,14839,14849,14859,14869,
14879,14889,14899,14912,14913,14914,14915,14916,14917,14918,14919,14923,14924,
14925,14926,14927,14928,14929,14934,14935,14936,14937,14938,14939,14945,14946,
14947,14948,14949,14956,14957,14958,14959,14967,14968,14969,14978,14979,14989,
14991,14992,14993,14994,14995,14996,14997,14998,14999,15129,15139,15149,15159,
15169,15179,15189,15199,15219,15229,15239,15249,15259,15269,15279,15289,15299,
15319,15329,15339,15349,15359,15369,15379,15389,15399,15419,15429,15439,15449,
15459,15469,15479,15489,15499,15519,15529,15539,15549,15559,15569,15579,15589,
15599,15619,15629,15639,15649,15659,15669,15679,15689,15699,15719,15729,15739,
15749,15759,15769,15779,15789,15799,15819,15829,15839,15849,15859,15869,15879,
15889,15899,15912,15913,15914,15915,15916,15917,15918,15919,15923,15924,15925,
15926,15927,15928,15929,15934,15935,15936,15937,15938,15939,15945,15946,15947,

170


15948,15949,15956,15957,15958,15959,15967,15968,15969,15978,15979,15989,15991,
15992,15993,15994,15995,15996,15997,15998,15999,16129,16139,16149,16159,16169,
16179,16189,16199,16219,16229,16239,16249,16259,16269,16279,16289,16299,16319,
16329,16339,16349,16359,16369,16379,16389,16399,16419,16429,16439,16449,16459,
16469,16479,16489,16499,16519,16529,16539,16549,16559,16569,16579,16589,16599,
16619,16629,16639,16649,16659,16669,16679,16689,16699,16719,16729,16739,16749,
16759,16769,16779,16789,16799,16819,16829,16839,16849,16859,16869,16879,16889,
16899,16912,16913,16914,16915,16916,16917,16918,16919,16923,16924,16925,16926,
16927,16928,16929,16934,16935,16936,16937,16938,16939,16945,16946,16947,16948,
16949,16956,16957,16958,16959,16967,16968,16969,16978,16979,16989,16991,16992,
16993,16994,16995,16996,16997,16998,16999,17129,17139,17149,17159,17169,17179,
17189,17199,17219,17229,17239,17249,17259,17269,17279,17289,17299,17319,17329,
17339,17349,17359,17369,17379,17389,17399,17419,17429,17439,17449,17459,17469,
17479,17489,17499,17519,17529,17539,17549,17559,17569,17579,17589,17599,17619,
17629,17639,17649,17659,17669,17679,17689,17699,17719,17729,17739,17749,17759,
17769,17779,17789,17799,17819,17829,17839,17849,17859,17869,17879,17889,17899,
17912,17913,17914,17915,17916,17917,17918,17919,17923,17924,17925,17926,17927,
17928,17929,17934,17935,17936,17937,17938,17939,17945,17946,17947,17948,17949,
17956,17957,17958,17959,17967,17968,17969,17978,17979,17989,17991,17992,17993,
17994,17995,17996,17997,17998,17999,18129,18139,18149,18159,18169,18179,18189,
18199,18219,18229,18239,18249,18259,18269,18279,18289,18299,18319,18329,18339,
18349,18359,18369,18379,18389,18399,18419,18429,18439,18449,18459,18469,18479,
18489,18499,18519,18529,18539,18549,18559,18569,18579,18589,18599,18619,18629,
18639,18649,18659,18669,18679,18689,18699,18719,18729,18739,18749,18759,18769,
18779,18789,18799,18819,18829,18839,18849,18859,18869,18879,18889,18899,18912,
18913,18914,18915,18916,18917,18918,18919,18923,18924,18925,18926,18927,18928,
18929,18934,18935,18936,18937,18938,18939,18945,18946,18947,18948,18949,18956,
18957,18958,18959,18967,18968,18969,18978,18979,18989,18991,18992,18993,18994,
18995,18996,18997,18998,18999,19129,19139,19149,19159,19169,19179,19189,19199,
19219,19229,19239,19249,19259,19269,19279,19289,19299,19319,19329,19339,19349,
19359,19369,19379,19389,19399,19419,19429,19439,19449,19459,19469,19479,19489,
19499,19519,19529,19539,19549,19559,19569,19579,19589,19599,19619,19629,19639,
19649,19659,19669,19679,19689,19699,19719,19729,19739,19749,19759,19769,19779,
19789,19799,19819,19829,19839,19849,19859,19869,19879,19889,19899,19912,19913,
19914,19915,19916,19917,19918,19919,19921,19922,19923,19924,19925,19926,19927,
19928,19929,19931,19932,19933,19934,19935,19936,19937,19938,19939,19941,19942,
19943,19944,19945,19946,19947,19948,19949,19951,19952,19953,19954,19955,19956,
19957,19958,19959,19961,19962,19963,19964,19965,19966,19967,19968,19969,19971,
19972,19973,19974,19975,19976,19977,19978,19979,19981,19982,19983,19984,19985,
19986,19987,19988,19989,19991,19992,19993,19994,19995,19996,19997,19998,19999}

171



{10019,10029,10039,10049,10059,10069,10079,10089,10099,10119,10129,10139,10149
,10159,10169,10179,10189,10199,10219,10229,10239,10249,10259,10269,10279,10289,
10299,10319,10329,10339,10349,10359,10369,10379,10389,10399,10419,10429,10439,
10449,10459,10469,10479,10489,10499,10519,10529,10539,10549,10559,10569,10579,
10589,10599,10619,10629,10639,10649,10659,10669,10679,10689,10699,10719,10729,
10739,10749,10759,10769,10779,10789,10799,10819,10829,10839,10849,10859,10869,
10879,10889,10899,10919,10929,10939,10949,10959,10969,10979,10989,10999,11019,
11119,11219,11319,11419,11519,11619,11719,11819,11919,12019,12029,12039,12049,
12059,12069,12079,12089,12099,12119,12219,12319,12419,12519,12619,12719,12819,
12919,13019,13029,13039,13049,13059,13069,13079,13089,13099,13119,13219,13319,
13419,13519,13619,13719,13819,13919,14019,14029,14039,14049,14059,14069,14079,
14089,14099,14119,14219,14319,14419,14519,14619,14719,14819,14919,15019,15029,
15039,15049,15059,15069,15079,15089,15099,15119,15219,15319,15419,15519,15619,
15719,15819,15919,16019,16029,16039,16049,16059,16069,16079,16089,16099,16119,
16219,16319,16419,16519,16619,16719,16819,16919,17019,17029,17039,17049,17059,
17069,17079,17089,17099,17119,17219,17319,17419,17519,17619,17719,17819,17919,
18019,18029,18039,18049,18059,18069,18079,18089,18099,18119,18219,18319,18419,
18519,18619,18719,18819,18919,19019,19029,19039,19049,19059,19069,19079,19089,
19099,19119,19219,19319,19419,19519,19619,19719,19819,19919,11029,11129,11229,
11329,11429,11529,11629,11729,11829,11929,12129,12229,12329,12429,12529,12629,
12729,12829,12929,13129,13229,13329,13429,13529,13629,13729,13829,13929,14129,
14229,14329,14429,14529,14629,14729,14829,14929,15129,15229,15329,15429,15529,
15629,15729,15829,15929,16129,16229,16329,16429,16529,16629,16729,16829,16929,
17129,17229,17329,17429,17529,17629,17729,17829,17929,18129,18229,18329,18429,
18529,18629,18729,18829,18929,19129,19229,19329,19429,19529,19629,19729,19829,
19929,11039,11139,11239,11339,11439,11539,11639,11739,11839,11939,12139,12239,
12339,12439,12539,12639,12739,12839,12939,13139,13239,13339,13439,13539,13639,
13739,13839,13939,14139,14239,14339,14439,14539,14639,14739,14839,14939,15139,
15239,15339,15439,15539,15639,15739,15839,15939,16139,16239,16339,16439,16539,
16639,16739,16839,16939,17139,17239,17339,17439,17539,17639,17739,17839,17939,
18139,18239,18339,18439,18539,18639,18739,18839,18939,19139,19239,19339,19439,
19539,19639,19739,19839,19939,11049,11149,11249,11349,11449,11549,11649,11749,
11849,11949,12149,12249,12349,12449,12549,12649,12749,12849,12949,13149,13249,
13349,13449,13549,13649,13749,13849,13949,14149,14249,14349,14449,14549,14649,
14749,14849,14949,15149,15249,15349,15449,15549,15649,15749,15849,15949,16149,
16249,16349,16449,16549,16649,16749,16849,16949,17149,17249,17349,17449,17549,
17649,17749,17849,17949,18149,18249,18349,18449,18549,18649,18749,18849,18949,
19149,19249,19349,19449,19549,19649,19749,19849,19949,11059,11159,11259,11359,
11459,11559,11659,11759,11859,11959,12159,12259,12359,12459,12559,12659,12759,
12859,12959,13159,13259,13359,13459,13559,13659,13759,13859,13959,14159,14259,
14359,14459,14559,14659,14759,14859,14959,15159,15259,15359,15459,15559,15659,

172


15759,15859,15959,16159,16259,16359,16459,16559,16659,16759,16859,16959,17159,
17259,17359,17459,17559,17659,17759,17859,17959,18159,18259,18359,18459,18559,
18659,18759,18859,18959,19159,19259,19359,19459,19559,19659,19759,19859,19959,
11069,11169,11269,11369,11469,11569,11669,11769,11869,11969,12169,12269,12369,
12469,12569,12669,12769,12869,12969,13169,13269,13369,13469,13569,13669,13769,
13869,13969,14169,14269,14369,14469,14569,14669,14769,14869,14969,15169,15269,
15369,15469,15569,15669,15769,15869,15969,16169,16269,16369,16469,16569,16669,
16769,16869,16969,17169,17269,17369,17469,17569,17669,17769,17869,17969,18169,
18269,18369,18469,18569,18669,18769,18869,18969,19169,19269,19369,19469,19569,
19669,19769,19869,19969,11079,11179,11279,11379,11479,11579,11679,11779,11879,
11979,12179,12279,12379,12479,12579,12679,12779,12879,12979,13179,13279,13379,
13479,13579,13679,13779,13879,13979,14179,14279,14379,14479,14579,14679,14779,
14879,14979,15179,15279,15379,15479,15579,15679,15779,15879,15979,16179,16279,
16379,16479,16579,16679,16779,16879,16979,17179,17279,17379,17479,17579,17679,
17779,17879,17979,18179,18279,18379,18479,18579,18679,18779,18879,18979,19179,
19279,19379,19479,19579,19679,19779,19879,19979,11089,11189,11289,11389,11489,
11589,11689,11789,11889,11989,12189,12289,12389,12489,12589,12689,12789,12889,
12989,13189,13289,13389,13489,13589,13689,13789,13889,13989,14189,14289,14389,
14489,14589,14689,14789,14889,14989,15189,15289,15389,15489,15589,15689,15789,
15889,15989,16189,16289,16389,16489,16589,16689,16789,16889,16989,17189,17289,
17389,17489,17589,17689,17789,17889,17989,18189,18289,18389,18489,18589,18689,
18789,18889,18989,19189,19289,19389,19489,19589,19689,19789,19889,19989,10091,
10092,10093,10094,10095,10096,10097,10098,10191,10192,10193,10194,10195,10196,
10197,10198,10291,10292,10293,10294,10295,10296,10297,10298,10391,10392,10393,
10394,10395,10396,10397,10398,10491,10492,10493,10494,10495,10496,10497,10498,
10591,10592,10593,10594,10595,10596,10597,10598,10691,10692,10693,10694,10695,
10696,10697,10698,10791,10792,10793,10794,10795,10796,10797,10798,10891,10892,
10893,10894,10895,10896,10897,10898,10991,10992,10993,10994,10995,10996,10997,
10998,11099,11199,11299,11399,11499,11599,11699,11799,11899,11991,11992,11993,
11994,11995,11996,11997,11998,11999,12091,12092,12093,12094,12095,12096,12097,
12098,12199,12299,12399,12499,12599,12699,12799,12899,12991,12992,12993,12994,
12995,12996,12997,12998,12999,13091,13092,13093,13094,13095,13096,13097,13098,
13199,13299,13399,13499,13599,13699,13799,13899,13991,13992,13993,13994,13995,
13996,13997,13998,13999,14091,14092,14093,14094,14095,14096,14097,14098,14199,
14299,14399,14499,14599,14699,14799,14899,14991,14992,14993,14994,14995,14996,
14997,14998,14999,15091,15092,15093,15094,15095,15096,15097,15098,15199,15299,
15399,15499,15599,15699,15799,15899,15991,15992,15993,15994,15995,15996,15997,
15998,15999,16091,16092,16093,16094,16095,16096,16097,16098,16199,16299,16399,
16499,16599,16699,16799,16899,16991,16992,16993,16994,16995,16996,16997,16998,
16999,17091,17092,17093,17094,17095,17096,17097,17098,17199,17299,17399,17499,
17599,17699,17799,17899,17991,17992,17993,17994,17995,17996,17997,17998,17999,
18091,18092,18093,18094,18095,18096,18097,18098,18199,18299,18399,18499,18599,
18699,18799,18899,18991,18992,18993,18994,18995,18996,18997,18998,18999,19091,
19092,19093,19094,19095,19096,19097,19098,19191,19192,19193,19194,19195,19196,
19197,19198,19199,19291,19292,19293,19294,19295,19296,19297,19298,19299,19391,
19392,19393,19394,19395,19396,19397,19398,19399,19491,19492,19493,19494,19495,

173


19496,19497,19498,19499,19591,19592,19593,19594,19595,19596,19597,19598,19599,
19691,19692,19693,19694,19695,19696,19697,19698,19699,19791,19792,19793,19794,
19795,19796,19797,19798,19799,19891,19892,19893,19894,19895,19896,19897,19898,
19899,19991,19992,19993,19994,19995,19996,19997,19998,19999}

{11019,11119,11129,11139,11149,11159,11169,11179,11189,11199,11219,11229,11239
,11249,11259,11269,11279,11289,11299,11319,11329,11339,11349,11359,11369,11379,
11389,11399,11419,11429,11439,11449,11459,11469,11479,11489,11499,11519,11529,
11539,11549,11559,11569,11579,11589,11599,11619,11629,11639,11649,11659,11669,
11679,11689,11699,11719,11729,11739,11749,11759,11769,11779,11789,11799,11819,
11829,11839,11849,11859,11869,11879,11889,11899,11919,11929,11939,11949,11959,
11969,11979,11989,11991,11992,11993,11994,11995,11996,11997,11998,11999,12019,
12119,12129,12139,12149,12159,12169,12179,12189,12199,12219,12229,12239,12249,
12259,12269,12279,12289,12299,12319,12329,12339,12349,12359,12369,12379,12389,
12399,12419,12429,12439,12449,12459,12469,12479,12489,12499,12519,12529,12539,
12549,12559,12569,12579,12589,12599,12619,12629,12639,12649,12659,12669,12679,
12689,12699,12719,12729,12739,12749,12759,12769,12779,12789,12799,12819,12829,
12839,12849,12859,12869,12879,12889,12899,12919,12929,12939,12949,12959,12969,
12979,12989,12991,12992,12993,12994,12995,12996,12997,12998,12999,13019,13119,
13129,13139,13149,13159,13169,13179,13189,13199,13219,13229,13239,13249,13259,
13269,13279,13289,13299,13319,13329,13339,13349,13359,13369,13379,13389,13399,
13419,13429,13439,13449,13459,13469,13479,13489,13499,13519,13529,13539,13549,
13559,13569,13579,13589,13599,13619,13629,13639,13649,13659,13669,13679,13689,
13699,13719,13729,13739,13749,13759,13769,13779,13789,13799,13819,13829,13839,
13849,13859,13869,13879,13889,13899,13919,13929,13939,13949,13959,13969,13979,
13989,13991,13992,13993,13994,13995,13996,13997,13998,13999,14019,14119,14129,
14139,14149,14159,14169,14179,14189,14199,14219,14229,14239,14249,14259,14269,
14279,14289,14299,14319,14329,14339,14349,14359,14369,14379,14389,14399,14419,
14429,14439,14449,14459,14469,14479,14489,14499,14519,14529,14539,14549,14559,
14569,14579,14589,14599,14619,14629,14639,14649,14659,14669,14679,14689,14699,
14719,14729,14739,14749,14759,14769,14779,14789,14799,14819,14829,14839,14849,
14859,14869,14879,14889,14899,14919,14929,14939,14949,14959,14969,14979,14989,
14991,14992,14993,14994,14995,14996,14997,14998,14999,15019,15119,15129,15139,
15149,15159,15169,15179,15189,15199,15219,15229,15239,15249,15259,15269,15279,
15289,15299,15319,15329,15339,15349,15359,15369,15379,15389,15399,15419,15429,
15439,15449,15459,15469,15479,15489,15499,15519,15529,15539,15549,15559,15569,
15579,15589,15599,15619,15629,15639,15649,15659,15669,15679,15689,15699,15719,
15729,15739,15749,15759,15769,15779,15789,15799,15819,15829,15839,15849,15859,
15869,15879,15889,15899,15919,15929,15939,15949,15959,15969,15979,15989,15991,
15992,15993,15994,15995,15996,15997,15998,15999,16019,16119,16129,16139,16149,

174


16159,16169,16179,16189,16199,16219,16229,16239,16249,16259,16269,16279,16289,
16299,16319,16329,16339,16349,16359,16369,16379,16389,16399,16419,16429,16439,
16449,16459,16469,16479,16489,16499,16519,16529,16539,16549,16559,16569,16579,
16589,16599,16619,16629,16639,16649,16659,16669,16679,16689,16699,16719,16729,
16739,16749,16759,16769,16779,16789,16799,16819,16829,16839,16849,16859,16869,
16879,16889,16899,16919,16929,16939,16949,16959,16969,16979,16989,16991,16992,
16993,16994,16995,16996,16997,16998,16999,17019,17119,17129,17139,17149,17159,
17169,17179,17189,17199,17219,17229,17239,17249,17259,17269,17279,17289,17299,
17319,17329,17339,17349,17359,17369,17379,17389,17399,17419,17429,17439,17449,
17459,17469,17479,17489,17499,17519,17529,17539,17549,17559,17569,17579,17589,
17599,17619,17629,17639,17649,17659,17669,17679,17689,17699,17719,17729,17739,
17749,17759,17769,17779,17789,17799,17819,17829,17839,17849,17859,17869,17879,
17889,17899,17919,17929,17939,17949,17959,17969,17979,17989,17991,17992,17993,
17994,17995,17996,17997,17998,17999,18019,18119,18129,18139,18149,18159,18169,
18179,18189,18199,18219,18229,18239,18249,18259,18269,18279,18289,18299,18319,
18329,18339,18349,18359,18369,18379,18389,18399,18419,18429,18439,18449,18459,
18469,18479,18489,18499,18519,18529,18539,18549,18559,18569,18579,18589,18599,
18619,18629,18639,18649,18659,18669,18679,18689,18699,18719,18729,18739,18749,
18759,18769,18779,18789,18799,18819,18829,18839,18849,18859,18869,18879,18889,
18899,18919,18929,18939,18949,18959,18969,18979,18989,18991,18992,18993,18994,
18995,18996,18997,18998,18999,19019,19119,19129,19139,19149,19159,19169,19179,
19189,19191,19192,19193,19194,19195,19196,19197,19198,19199,19219,19229,19239,
19249,19259,19269,19279,19289,19291,19292,19293,19294,19295,19296,19297,19298,
19299,19319,19329,19339,19349,19359,19369,19379,19389,19391,19392,19393,19394,
19395,19396,19397,19398,19399,19419,19429,19439,19449,19459,19469,19479,19489,
19491,19492,19493,19494,19495,19496,19497,19498,19499,19519,19529,19539,19549,
19559,19569,19579,19589,19591,19592,19593,19594,19595,19596,19597,19598,19599,
19619,19629,19639,19649,19659,19669,19679,19689,19691,19692,19693,19694,19695,
19696,19697,19698,19699,19719,19729,19739,19749,19759,19769,19779,19789,19791,
19792,19793,19794,19795,19796,19797,19798,19799,19819,19829,19839,19849,19859,
19869,19879,19889,19891,19892,19893,19894,19895,19896,19897,19898,19899,19919,
19929,19939,19949,19959,19969,19979,19989,19991,19992,19993,19994,19995,19996,
19997,19998,19999}

Números de 5 cifras que se pueden expresar como suma de dos primos dismales

La unión de los D1, D2 , D3 y D4 nos dará los números de 5 cifras que se pueden
expresar como suma de dos primos dismales.

175


{11019,11119,11129,11139,11149,11159,11169,11179,11189,11191,11192,11193,11194
,11195,11196,11197,11198,11199,11219,11229,11239,11249,11259,11269,11279,11289,
11291,11292,11293,11294,11295,11296,11297,11298,11299,11319,11329,11339,11349,
11359,11369,11379,11389,11391,11392,11393,11394,11395,11396,11397,11398,11399,
11419,11429,11439,11449,11459,11469,11479,11489,11491,11492,11493,11494,11495,
11496,11497,11498,11499,11519,11529,11539,11549,11559,11569,11579,11589,11591,
11592,11593,11594,11595,11596,11597,11598,11599,11619,11629,11639,11649,11659,
11669,11679,11689,11691,11692,11693,11694,11695,11696,11697,11698,11699,11719,
11729,11739,11749,11759,11769,11779,11789,11791,11792,11793,11794,11795,11796,
11797,11798,11799,11819,11829,11839,11849,11859,11869,11879,11889,11891,11892,
11893,11894,11895,11896,11897,11898,11899,11911,11912,11913,11914,11915,11916,
11917,11918,11919,11921,11922,11923,11924,11925,11926,11927,11928,11929,11931,
11932,11933,11934,11935,11936,11937,11938,11939,11941,11942,11943,11944,11945,
11946,11947,11948,11949,11951,11952,11953,11954,11955,11956,11957,11958,11959,
11961,11962,11963,11964,11965,11966,11967,11968,11969,11971,11972,11973,11974,
11975,11976,11977,11978,11979,11981,11982,11983,11984,11985,11986,11987,11988,
11989,11991,11992,11993,11994,11995,11996,11997,11998,11999,12019,12119,12129,
12139,12149,12159,12169,12179,12189,12191,12192,12193,12194,12195,12196,12197,
12198,12199,12219,12229,12239,12249,12259,12269,12279,12289,12291,12292,12293,
12294,12295,12296,12297,12298,12299,12319,12329,12339,12349,12359,12369,12379,
12389,12391,12392,12393,12394,12395,12396,12397,12398,12399,12419,12429,12439,
12449,12459,12469,12479,12489,12491,12492,12493,12494,12495,12496,12497,12498,
12499,12519,12529,12539,12549,12559,12569,12579,12589,12591,12592,12593,12594,
12595,12596,12597,12598,12599,12619,12629,12639,12649,12659,12669,12679,12689,
12691,12692,12693,12694,12695,12696,12697,12698,12699,12719,12729,12739,12749,
12759,12769,12779,12789,12791,12792,12793,12794,12795,12796,12797,12798,12799,
12819,12829,12839,12849,12859,12869,12879,12889,12891,12892,12893,12894,12895,
12896,12897,12898,12899,12911,12912,12913,12914,12915,12916,12917,12918,12919,
12921,12922,12923,12924,12925,12926,12927,12928,12929,12931,12932,12933,12934,
12935,12936,12937,12938,12939,12941,12942,12943,12944,12945,12946,12947,12948,
12949,12951,12952,12953,12954,12955,12956,12957,12958,12959,12961,12962,12963,
12964,12965,12966,12967,12968,12969,12971,12972,12973,12974,12975,12976,12977,
12978,12979,12981,12982,12983,12984,12985,12986,12987,12988,12989,12991,12992,
12993,12994,12995,12996,12997,12998,12999,13019,13119,13129,13139,13149,13159,
13169,13179,13189,13191,13192,13193,13194,13195,13196,13197,13198,13199,13219,
13229,13239,13249,13259,13269,13279,13289,13291,13292,13293,13294,13295,13296,
13297,13298,13299,13319,13329,13339,13349,13359,13369,13379,13389,13391,13392,
13393,13394,13395,13396,13397,13398,13399,13419,13429,13439,13449,13459,13469,
13479,13489,13491,13492,13493,13494,13495,13496,13497,13498,13499,13519,13529,
13539,13549,13559,13569,13579,13589,13591,13592,13593,13594,13595,13596,13597,
13598,13599,13619,13629,13639,13649,13659,13669,13679,13689,13691,13692,13693,
13694,13695,13696,13697,13698,13699,13719,13729,13739,13749,13759,13769,13779,
13789,13791,13792,13793,13794,13795,13796,13797,13798,13799,13819,13829,13839,
13849,13859,13869,13879,13889,13891,13892,13893,13894,13895,13896,13897,13898,

176

La Conjetura de Goldbach en la Aritmética Dismal

La Conjetura de Goldbach en la Aritmética Dismal

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La Conjetura de Goldbach en la Aritmética Dismal