Números Primos (la descomposición numérica de Sergio Aschero)

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En este trabajo se desarrolla un modelo diferente para descubrir los números primos a través de una descomposición numérica.

Transcript of Números Primos (la descomposición numérica de Sergio Aschero)

Nmeros Primos: la descomposicin numrica de Sergio AscheroPaso 1: Todo nmero puede descomponerse en otro u otros nmeros equivalentes que tienen un lmite (corte) donde se produce la inversin. El anlisis se realiza con los nmeros anteriores al corte, ya que los posteriores repiten el mismo esquema numrico invertido. 7 = 1/6, 2/5, 3/4 - 4/3, 5/2, 6/1 Es a travs de ese procedimiento que se pueden descubrir los nmeros compuestos y los primos sin utilizar obligatoriamente la divisin, que en este anlisis puede reemplazarse por la suma. Si la suma consecutiva del intervalo que determinan los dos nmeros menores elegidos concluye con el resultado del nmero inicial, el nmero es compuesto y si es discordante el nmero es primo. Queda fuera del anlisis el intervalo 1 por no diferenciar ambas caractersticas numricas. 9 = 1/8, 2/7, 3/6, 4/5 (3/6 = + 3) (3, 6, 9) (el nmero es compuesto) 5 = 1/4, 2/3 (el nmero es primo) Paso 2: 3:2 = 1,5 (el decimal se descarta) y el primer corte (antes de la inversin es 1/2) (nmeros sucesivos) 5:2 = 2,5 (el decimal se descarta) y el primer corte (lmite de la descomposicin previa a la inversin) es 2/3) 51:2 = 25,5 (el decimal se descarta) y el primer corte (lmite de la descomposicin previa a la inversin) es 25/26) Paso 3: 143 = (143:2 = 71,5) (65/78), 66/77, 67/76, 68/75, 69/74, 70/73, 71/72 (66, 77, 88, 99, 110, 121, 132, 143) (66/77 = + 11) (suma) 71/72 = 1, 70/73 = 3, 69/74 = 5, 68/75 = 7, 67/76 = 9 66/77 = 11 (65/78) = 13 (divisin) Se puede dividir 143 por 11 y por 13 y por supuesto por 143 Siempre se debe tomar como referencia el nmero menor que en este caso es el 11 y es concordante con el nmero de la suma)

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3 = 1/2 (descomposicin) (2/1) (inversin) (primo) 5 = 1/4, 2/3 (descomposicin) (3/2, 4/1) (inversin) (primo) 7 = 1/6, 2/5, 3/4 (descomposicin) (4/3, 5/2, 6/1) (inversin) (primo) 9 = 1/8, 2/7, 3/6, 4/5 (descomposicin) (5/4, 6/3, 7/2, 8/1) (inversin) (3, 6, 9) (compuesto) (3/6 = + 3) (3) 11 = 1/10, 2/9, 3/8, 4/7, 5/6 (primo) 13 = 1/12, 2/11, 3/10, 4/9, 5/8, 6/7 (primo) 15 = 1/14, 2/13, 3/12, 4/11, (5/10), 6/9, 7/8 (6/9 = + 3) (3) a) (5/10 = + 5) (5) b) 17 = 1/16, 2/15, 3/14, 4/13, 5/12, 6/11, 7/10, 8/9 (primo) 19 = 1/18, 2/17, 3/16, 4/15, 5/14, 6/13, 7/12, 8/11, 9/10 (primo) 21 = 1/20, 2/19, 3/18, 4/17, 5/16, 6/15, (7/14), 8/13, 9/12, 10/11 (9/12 = + 3) (3) a) 23 = 1/22, 2/21, 3/20, 4/19, 5/18, 6/17, 7/16, 8/15, 9/14, 10/13, 11/12 25 = 1/24, 2/23, 3/22, 4/21, 5/20, 6/19, 7/18, 8/17, 9/16, 10/15, 11/14, 12/13 (10/15 = + 5) (5) 27 = 1/26, 2/25, 3/24, 4/23, 5/22, 6/21, 7/20, 8/19, (9/18), 10/17, 11/16, 12/15, 13/14 (12/15 = + 3) (3) 29 = 1/28, 2/27, 3/26, 4/25, 5/24, 6/23, 7/22, 8/21, 9/20, 10/19, 11/18, 12/17, 13/16, 14/15 31 = 1/30, 2/29, 3/28, 4/27, 5/26, 6/25, 7/24, 8/23, 9/22, 10/21, 11/20, 12/19, 13/18, 14/17, 15/16 33 = 1/32, 2/31, 3/30, 4/29, 5/28, 6/27, 7/26, 8/25, 9/24, 10/23, (11/22), 12/21, 13/20, 14/19, 15/18, 16/17 (15/18 = + 3) (3) 35 = 1/34, 2/33, 3/32, 4/31, 5/30, 6/29, 7/28, 8/27, 9/26, 10/25, 11/24, 12/23, 13/22, (14/21), 15/20, 16/19, 17/18 (15/20 = + 5) (5) 37 = 1/36, 2/35, 3/34, 4/33, 5/32, 6/31, 7/30, 8/29, 9/28, 10/27, 11/26, 12/25, 13/24, 14/23, 15/22, 16/21, 17/20, 18/19

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39 = 1/38, 2/37, 3/36, 4/35, 5/34, 6/33, 7/32, 8/31, 9/30, 10/29, 11/28, 12/27, (13/26), 14/25, 15/24, 16/23, 17/22, 18/21, 19/20 (18/21 = + 3) (3) 41 = 1/40, 2/39, 3/38, 4/37, 5/36, 6/35, 7/34, 8/33, 9/32, 10/31, 11/30, 12/29, 13/28, 14/27, 15/26, 16/25, 17/24, 18/23, 19/22, 20/21 43 = 1/42, 2/41, 3/40, 4/39, 5/38, 6/37, 7/36, 8/35, 9/34, 10/33, 11/32, 12/31, 13/30, 14/29, 15/28, 16/27, 17/26, 18/25, 19/24, 20/23, 21/22 45 = 1/44, 2/43, 3/42, 4/41, 5/40, 6/39, 7/38, 8/37, 9/36, 10/35, 11/34, 12/33, 13/32, 14/31, (15/30), 16/29, 17/28, (18/27), 19/26, (20/25), 21/24, 22/23 (21/24 = + 3) (3) 47 = 1/46, 2/45, 3/44, 4/43, 5/42, 6/41, 7/40, 8/39, 9/38, 10/37, 11/36, 12/35, 13/34, 14/33, 15/32, 16/31, 17/30, 18/29, 19/28, 20/27, 21/26, 22/25, 23/24 49 = 1/48, 2/47, 3/46, 4/45, 5/44, 6/43, 7/42, 8/41, 9/40, 10/39, 11/38, 12/37, 13/36, 14/35, 15/34, 16/33, 17/32, 18/31, 19/30, 20/29, 21/28, 22/27, 23/26, 24/25 (21/28 = + 7) (7) 51 = 1/50, 2/49, 3/48, 4/47, 5/46, 6/45, 7/44, 8/43, 9/42, 10/41, 11/40, 12/39, 13/38, 14/37, 15/36, 16/35, (17/34), 18/33, 19/32, 20/31, 21/30, 22/29, 23/28, 24/27, 25/26 (24/27 = + 3) (3) 53 = 1/52, 2/51, 3/50, 4/49, 5/48, 6/47, 7/46, 8/45, 9/44, 10/43, 11/42, 12/41, 13/40, 14/39, 15/38, 16/37, 17/36, 18/35, 19/34, 20/33, 21/32, 22/31, 23/30,24/29, 25/28, 26/27 55 = 1/54, 2/53, 3/52, 4/51, 5/50, 6/49, 7/48, 8/47, 9/46, 10/45, 11/44, 12/43, 13/42, 14/41, 15/40, 16/39, 17/38, 18/37, 19/36, 20/35, 21/34, (22/33), 23/32, 24/31, 25/30, 26/29, 27/28 (25/30 = + 5) (5) 57 = 1/56, 2/55, 3/54, 4/53, 5/52, 6/51, 7/50, 8/49, 9/48, 10/47, 11/46, 12/45, 13/44, 14/43, 15/42, 16/41, 17/40, 18/39, (19/38), 20/37, 21/36 , 22/35, 23/34, 24/33, 25/32, 26/31, 27/30, 28/29 (27/30 = + 3) (3) 59 = 1/58, 2/57, 3/56, 4/55, 5/54, 6/53, 7/52, 8/51, 9/50, 10/49, 11/48, 12/47, 13/46, 14/45, 15/44, 16/43, 17/42, 18/41, 19/40, 20/39, 21/38, 22/37, 23/36, 24/35, 25/34, 26/33, 27/32, 28/31, 29/30 61 = 1/60, 2/59, 3/58, 4/57, 5/56, 6/55, 7/54, 8/53, 9/52, 10/51, 11/50, 12/49, 13/48, 14/47, 15/46, 16/45, 17/44, 18/43, 19/42, 20/41, 21/40, 22/39, 23/38, 24/37, 25/36, 26/35, 27/34, 28/33, 29/32, 30/31 63 = (63:2 = 31,5) 30/33, 31/32 (30/33 = + 3) (3)3

65 = (65:2 = 32,5) 30/35, 31/34, 32/33 (30/35 = + 5) (5) 67 = (67:2 = 33,5) 31/36, 32/35, 33/34 69 = (69:2 = 34,5) 33/36, 34/35 (33/36 = + 3) (3) 71 = (71:2 = 35,5) 33/38, 34/37, 35/36 73 = (73:2 = 36,5) 34/39, 35/38, 36/37 75 = (75:2 = 37,5) 35/40, 36/39, 37/38 (35/40 = + 5) (5) 77 = (77:2 = 38,5) 35/42, 36/41, 37/40, 38/39 (35/42 = + 7) (7) 79 = (79:2 = 39,5) 36/43, 37/42, 38/41, 39/40 81 = (81:2 = 40,5) 39/42, 40/41 (39/42 = + 3) (3) 83 = (83:2 = 41,5) 38/45, 39/44, 40/43, 41/42 85 = (85:2 = 42,5) 40/45, 41/44, 42/43 (40/45 = + 5) (5) 87 = (87:2 = 43,5) 42/45, 43/44 (42/45 = + 3) (3) 89 = (89:2 = 44,5) 41/48, 42/47, 43/46, 44/45 91 = (91:2 = 45,5) 42/49, 43/48, 44/47, 45/46 (42/49 = + 7) (7) 93 = (93:2 = 46,5) 45/48, 46/47 (45/48 = + 3) (3) 95 = (95:2 = 47,5) 45/50, 46/49, 47/48 (45/50 = + 5) (5) 97 = (97:2 = 48,5) 45/52, 46/51, 47/50, 48/49 99 = (99:2 = 49,5) 48/51, 49/50 (48/51 = + 3) (3) 101 = 50/51 103 = 51/52 105 = 50/55, 51/54, 52/53 (50/55 = + 5) (5) 107 = 53/54 109 = 54/55 111 = 54/57, 55/56 ((54/57 = + 3) (3)4

113 = 56/57 115 = 55/60, 56/59, 57/58 (55/60 = + 5) (5) 117 = 57/60, 58/59 ((57/60 = + 3) (3) 119 = 56/63, 57/62, 58/61, 59/60 (56/63 = + 7) (7) 121 = 55/66, 56/65, 57/64, 58/63, 59/62, 60/61 (55/66 = + 11) (11) 123 = 60/63, 61/62 (60/63 = + 3) (3) 125 = 60/65, 61/64, 62/63 (60/65 = + 5) (5) 127 = 63/64 129 = 63/66, 64/65 (63/66 = + 3) (3) 131 = 65/66 133 = 63/70, 64/69, 65/68, 66/67 (63/70 = + 7) (7) 135 = (135:2 = 67,5) 65/70, 66/69, 67/68 (65/70 = + 5) (5) 137 = 68/69 139 = 69/70 141 = 69/72, 70/71 ((69/72 = + 3) (3) 143 = 66/77, 67/76, 68/75, 69/74, 70/73, 71/72 (66/77 = + 11) (11) 145 = 70/75, 71/74, 72/73 (70/75 = + 5) (5) 147 = 72/75, 73/74 (72/75 = + 3) (3) 149 = 74/75 151 = 75/76 153 = 75/78, 76/77 (72/78 = + 3) (3) 155 = 75/80, 76/79, 77/78 (75/80 = + 5) (5) 157 = 78/795

159 = 78/81, 79/80 (78/81 = + 3) (3) 161 = 77/84, 78/83, 79/82, 80/81 (77/84 = + 7) (7)

163 = 81/82 165 = 81/84, 82/83 (81/84 = + 3) (3) 167 = 83/84 169 = 78/91, 79/90, 80/89, 81/88, 82/87, 83/86, 84/85 (78/91 = + 13) (13) 171 = 84/87, 85/86 (84/87 = + 3) (3) 173 = 86/87 175 = 85/90, 86/89, 87/88 (85/90 = + 5) (5) 177 = 87/90, 88/89 (87/90 = + 3) (3) 179 = 89/90 181 = 90/91 183 = 90/93, 91/92 (90/93 = + 3) (3) 185 = 90/95, 91/94, 92/93 (90/95 = + 5) (5) 187 = 88/99, 89/98, 90/97, 91/96, 92/95, 93/94 (88/99 = + 11) (11) 189 = 93/96, 94/95 (93/96 = + 3) (3) 191 = 95/96 193 = 96/97 195 = 96/99, 97/98 (96/99 = + 3) (3) 197 = 98/99 199 = 99/100 201 = 99/102, 100/101 (99/102 = + 3) (3) 203 = 98/105, 99/104, 100/103, 101/102 (98/105 = + 7) (7)6

205 = 100/105,101/104, 102/103 (100/105 = + 5) (5) 207 = 102/105, 103/104 (102/105 = + 3) (3) 209 = 99/110, 100/109, 101/108, 102/107, 103/106, 104/105 (99/101 = + 11) (11) 211 = 105/106 213 = 105/108, 106/107 (105/108 = + 3) (3) 215 = 105/110,106/109, 107/108 (105/110 = + 5) (5) 217 = 105/112,106/111,107/110, 108/109 (105/112 = + 7) (7) 219 = 108/111, 109/110 (108/111 = + 3) (3) 221 = 104/117, 105/116, 106/115, 107/114, 108/113, 109/112, 110/111 (104/117 = + 13) (13) 223 = 111/112 225 = 111/114, 112/113 (111/114 = + 3) (3) 227 = 113/114 229 = 114/115 231 = 114/117, 115/116 (114/117 = + 3) (3) 233 = 116/117 235 = 115/120,116/119, 117/118 (115/120 = + 5) (5) 237 = 117/120, 118/119 (117/120 = + 3) (3) 239 = 119/120 241 = 120/121 243 = 120/123, 121/122 (120/123 = + 3) (3) 245 = 120/125,121/124, 122/123 (120/125 = + 5) (5) 247 = 117/130, 118/129, 119/128, 120/127, 121/126, 122/125, 123/124 (117/130 = + 13) (13)7

249 = 123/126, 124/125 (123/126 = + 3) (3) 251 = 125/126 253 = 121/132, 122/131, 123/130, 124/129, 125/128, 126/127 (121/132 = + 11) (11) 255 = 126/129, 127/128 (126/129 = + 3) (3) 257 = 128/129 259 = 126/133,127/132,128/131, 129/130 (126/133 = + 7) (7) 261 = 129/132, 130/131 (129/132 = + 3) (3) 263 = 131/132 265 = 130/135,131/134, 132/133 (130/135 = + 5) (5) 267 = 132/135, 133/134 (132/135 = + 3) (3) 269 = 134/135 271 = 135/136 273 = 135/138, 136/137 (135/138 = + 3) (3) 275 = 135/140,136/139, 137/138 (135/140 = + 5) (5) 277 = 138/139 279 = 138/141, 139/140 (138/141 = + 3) (3) 281 = 140/141 283 = 141/142 285 = 141/144, 142/143 (141/144 = + 3) (3) 287 = 140/147,141/146,142/145, 143/144 (140/147 = + 7) (7) 289 = 136/153, 137/152, 138/151, 139/150, 140/149, 141/148, 142/147, 143/146, 144/145 (136/153 = + 17) (17) 291 = 144/147, 145/146 (144/147 = + 3) (3)8

293 = 146/147 295 = 145/150,146/149, 147/148 (145/150 = + 5) (5) 297 = 147/150, 148/149 (147/150 = + 3) (3) 299 = 143/156, 144/155, 145/154, 146/153, 147/152, 148/151, 149/150 (143/156 = + 13) (13) 301 = 147/154,148/153,149/152, 150/151 (147/154 = + 7) (7) 303 = 150/153, 151/152 (150/153 = + 3) (3) 305 = 150/155,151/154, 152/153 (150/155 = + 5) (5) 307 = 153/154 309 = 153/156, 154/155 (153/156 = + 3) (3) 311 = 155/156 313 = 156/157 315 = 156/159, 157/158 (156/159 = + 3) (3) 317 = 158/159 319 = 154/165, 155/164, 156/163, 157/162, 158/161, 159/160 (154/165 = + 11) (11) 321 = 159/162, 160/161 (159/162 = + 3) (3) 323 = 153/170, 154/169, 155/168, 156/167, 157/166, 158/165, 159/164, 160/163, 161/162 (153/170 = + 17) (17) 325 = 160/165,161/164, 162/163 (160/165 = + 5) (5) 327 = 162/165, 163/164 (162/165 = + 3) (3) 329 = 161/168,162/167,163/166, 164/165 (161/168 = + 7) (7) 331 = 165/166 333 = 165/168, 166/167 (165/168 = + 3) (3) 335 = 165/170,166/169, 167/168 (165/170 = + 5) (5) 337 = 168/1699

339 = 168/171, 169/170 (168/171 = + 3) (3) 341 = 165/176, 166/175, 167/174, 168/173, 169/172, 170/171 (165/176 = + 11) (11) 343 = 168/175,169/174,170/173, 171/172 (168/175 = + 7) (7) 345 = 171/174, 172/173 (171/174 = + 3) (3) 347 = 173/174 349 = 174/175 351 = 174/177, 175/176 (174/177 = + 3) (3) 353 = 175/176 355 = 174/179,175/178, 176/177 (174/179 = + 5) (5) 357 = 176/179, 177/178 (176/179 = + 3) (3) 359 = 178/179 361 = 170/189, 171/188, 172/187, 173/186, 174/185, 175/184, 176/183, 177/182, 178/181, 179/180 (170/189 = + 19) (19) 363 = 179/182, 180/181 (179/182 = + 3) (3) 365 = 179/184,180/183, 181/182 (179/184 = + 5) (5) 367 = 182/183 369 = 182/185, 183/184 (182/185 = + 3) (3) 371 = 181/188,182/187,183/186, 184/185 (181/188 = + 7) (7) 373 = 185/186 375 = 182/185, 183/184 (182/185 = + 3) (3) 377 = 178/191, 179/190, 180/189, 181/188, 182/187, 183/186, 184/185 (178/191 = + 13) (13) 379 = 185/18610

381 = 185/188, 186/187 (185/188 = + 3) (3) 383 = 187/188 385 = 186/191,187/190, 188/189 (186/191 = + 5) (5) 387 = 188/191, 189/190 (188/191 = + 3) (3) 389 = 190/191 391 = 183/200, 184/199, 185/198, 186/197, 187/196, 188/195, 189/194, 190/193, 191/192 (183/200 = + 17) (17) 393 = 191/194, 192/193 (191/194 = + 3) (3) 395 = 191/196,192/195, 193/194 (191/196 = + 5) (5) 397 = 194/195 399 = 194/197, 195/196 (194/197 = + 3) (3) 401 = 196/197 403 = 191/204, 192/203, 193/202, 194/201, 195/200, 196/199, 197/198 (191/204 = + 13) (13) 405 = 197/200, 198/199 (197/200 = + 3) (3) 407 = 193/204, 194/203, 195/202, 196/201, 197/200, 198/199 (193/204 = + 11) 409 = 199/200 411 = 199/202, 200/201 (199/202 = + 3) (3) 413 = 198/205,199/204, 200/203, 201/202 (198/205 = + 7) (7) 415 = 200/205, 201/204, 202/203 (200/205 = + 5) (5) 417 = 202/205, 203/204 (202/205 = + 3) (3) 419 = 204/205 421 = 205/206

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423 = 205/208, 206/207 (205/208 = + 3) (3) 425 = 205/210, 206/209, 207/208 (205/210 = + 5) (5) 427 = 205/212, 206/211, 207/210, 208/209 (205/212 = + 7) (7) 429 = 208/211, 209/210 (208/211 = + 3) (3) 431 = 210/211 433 = 211/212 435 = 211/214, 212/213 (211/214 = + 3) (3) 437 = 204/223, 205/222, 206/221, 207/220, 208/219, 209/218, 210/217, 211/216, 212/215, 213/214 (204/223 = + 19) (19) 439 = 214/215 441 = 214/217, 215/216 (214/217 = + 3) (3) 443 = 216/217 445 = 215/220, 216/219, 217/218 (215/220 = + 5) (5) 447 = 217/220, 218/219 (217/220 = + 3) (3) 449 = 219/220 451 = 115/226, 116/225, 117/224, 118/223, 119/222, 220/221 (115/226 = + 11) 453 = 119/222, 220/221 (119/222 = + 3) (3) 455 = 219/224, 220/223, 221/222 (219/224 = + 5) (5) 457 = 222/223 459 = 222/225, 223/224 (222/225 = + 3) (3) 461 = 224/225 463 = 225/226 465 = 225/228, 226/227 (225/228 = + 3) (3)12

467 = 227/228 469 = 225/232, 226/231, 227/230, 228/229 (225/232 = + 7) (7) 471 = 228/231, 229/230 (228/231 = + 3) (3) 473 = 225/236, 226/235, 227/234, 228/233, 229/232, 230/231 (225/236 = + 11) (11) 475 = 229/234, 230/233, 231/232 (229/234 = + 5) (5) 477 = 231/234, 232/233 (231/234 = + 3) (3) 479 = 233/234 481 = 228/241, 229/240, 230/239, 231/238, 232/237, 233/236, 234/235 (228/241 = + 13) (13) 483 = 234/237, 235/236 (234/237 = + 3) (3) 485 = 234/239, 235/238, 236/237 (234/239 = + 5) (5) 487 = 237/238 489 = 237/240, 238/239 (237/240 = + 3) (3) 491 = 239/240 493 = 232/249, 233/248, 234/247, 235/246, 236/245, 237/244, 238/243, 239/242, 240/241 (232/249 = + 17) (17) 495 = 240/243, 241/242 (240/243 = + 3) (3) 497 = 239/246, 240/245, 241/244, 242/243 (239/246 = + 7) (7) 499 = 243/244 501 = 243/246, 244/245 (243/246 = + 3) (3) 503 = 245/246 505 = 244/249, 245/248, 246/247 (244/249 = + 5) (5) 507 = 246/249, 247/248 (246/249 = + 3) (3) 509 = 248/24913

511 = 246/253, 247/252, 248/251, 249/250 (246/253 = + 7) (7) 513 = 249/252, 250/251 (249/252 = + 3) (3) 515 = 249/254, 250/253, 251/252 (249/254 = + 5) (5) 517 = 247/258, 248/257, 249/256, 250/255, 251/254, 252/253 (247/258 = + 11) (11) 519 = 252/255, 253/254 (252/255 = + 3) (3) 521 = 254/255 523 = 255/256 525 = 255/258, 256/257 (255/258 = + 3) (3) 527 = 248/265, 249/264, 250/263, 251/262, 252/261, 253/260, 254/259, 255/258, 256/257 (248/265 = + 17) (17) 529 = 246/269, 247/268, 248/267, 249/266, 250/265, 251/264, 252/263, 253/262, 254/261, 255/260, 256/259, 257/258 (246/269 = + 23) (23) 531 = 257/260, 258/259 (257/260 = + 3) (3) 533 = 253/266, 254/265, 255/264, 256/263, 257/262, 258/261, 259/260 (253/266 = + 13) (13) 535 = 258/263, 259/262, 260/261 (258/263 = + 5) (5) 537 = 260/263, 261/262 (257/260 = + 3) (3) 539 = 259/266, 260/265, 261/264, 262/263 (259/266 = + 7) (7) 541 = 263/264 543 = 263/266, 264/265 (263/266 = + 3) (3) 545 = 263/268, 264/267, 265/266 (263/268 = + 5) (5) 547 = 266/267 549 = 266/269, 267/268 (266/269 = + 3) (3) 551 = 259/278, 260/277, 261/276, 262/275, 263/274, 264/273, 265/272, 266/271, 267/270, 268/269 (259/278 = + 19) (19)14

553 = 266/273, 267/272, 268/271, 269/270 (266/278 = + 7) (7) 555 = 269/272, 270/271 (269/272 = + 3) (3) 557 = 271/272 559 = 266/279, 267/278, 268/277, 269/276, 270/275, 271/274, 272/273 (266/279 = + 13) (13) 561 = 272/275, 273/274 (272/275 = + 3) (3) 563 = 274/275 565 = 273/278, 274/277, 275/276 (273/278 = + 5) (5) 567 = 275/278, 276/277 (275/278 = + 3) (3) 569 = 277/278 571 = 278/279 573 = 278/281, 279/280 (278/281 = + 3) (3) 575 = 278/283, 279/282, 280/281 (278/283 = + 5) (5) 577 = 281/282 579 = 281/284, 282/283 (281/284 = + 3) (3) 581 = 280/287, 281/286, 282/285, 283/284 (280/287 = + 7) (7) 583 = 279/290, 280/289, 281/288, 282/287, 283/286, 284/285 (279/290 = + 11) (11) 585 = 284/287, 285/286 (284/287 = + 3) (3) 587 = 286/287 589 = 278/297, 279/296, 280/295, 281/294, 282/293, 283/292, 284/291, 285/290, 286/289, 287/288 (278/297 = + 19) (19) 591 = 287/290, 288/289 (287/290 = + 3) (3) 593 = 289/290 595 = 288/293, 289/292, 290/291 (288/293 = + 5) (5)15

597 = 290/293, 291/292 (290/293 = + 3) (3) 599 = 292/293 601 = 293/294 603 = 293/296, 294/295 (293/296 = + 3) (3) 605 = 293/298, 294/297, 295/296 (293/298 = + 5) (5) 607 = 296/297 609 = 296/299, 297/298 (296/299 = + 3) (3) 611 = 292/305, 293/304, 294/303, 295/302, 296/301, 297/300, 298/299 (292/305 = + 13) (13) 613 = 299/300 615 = 299/302, 300/301 (299/302 = + 3) (3) 617 = 301/302 619 = 302/303 621 = 302/305, 303/304 (302/305 = + 3) (3) 623 = 301/308, 302/307, 303/306, 304/305 (301/308 = + 7) (7) 625 = 303/308, 304/307, 305/306 (303/308 = + 5) (5) 627 = 305/308, 306/307 (305/308 = + 3) (3) 629 = 299/316, 300/315, 301/314, 302/313, 303/312, 304/311, 305/310, 306/309, 307/308 (299/316 = + 17) (17) 631 = 308/309 633 = 310/311, 309/310 (310/311 = + 3) (3) 635 = 308/313, 309/312, 310/311 (308/313 = + 5) (5) 637 = 308/315, 309/314, 310/313, 311/312 (308/315 = + 7) (7) 639 = 311/314, 312/313 (311/314 = + 3) (3) 641 = 313/31416

643 = 314/315 645 = 314/317, 315/316 (314/317 = + 3) (3) 647 = 316/317 649 = 312/323, 313/322, 314/321, 315/320, 316/319, 317/318 (312/323 = + 11) (11) 651 = 317/320, 318/319 (317/320 = + 3) (3) 653 = 319/320 655 = 318/323, 319/322, 320/321 (318/323 = + 5) (5) 657 = 320/323, 321/322 (320/323 = + 3) (3) 659 = 322/323 661 = 323/324 663 = 323/326, 324/325 (323/326 = + 3) (3) 665 = 323/328, 324/327, 325/326 (323/328 = + 5) (5) 667 = 315/338, 316/337, 317/336, 318/335, 319/334, 320/333, 321/332, 322/331, 323/330, 324/329, 325/328, 326/327 (315/338 = + 23) (23) 669 = 326/329, 327/328 (326/329 = + 3) (3) 671 = 323/334, 324/333, 325/332, 326/331, 327/330, 328/329 (312/323 = + 11) (11) 673 = 329/330 675 = 329/332, 330/331 (329/332 = + 3) (3) 677 = 331/332 679 = 329/336, 330/335, 331/334, 332/333 (329/336 = + 7) (7) 681 = 332/335, 333/334 (332/335 = + 3) (3) 683 = 334/335 685 = 333/338, 334/337, 335/336 (333/338 = + 5) (5)17

687 = 335/338, 336/337 (335/338 = + 3) (3) 689 = 328/347, 329/346, 330/345, 331/344, 332/343, 333/342, 334/341, 335/340, 336/339, 337/338 (328/347 = + 19) (19) 691 = 338/339 693 = 338/341, 339/340 (338/341 = + 3) (3) 695 = 338/343, 339/342, 340/341 (338/343 = + 5) (5) 697 = 333/350, 334/349, 335/348, 336/347, 337/346, 338/345, 339/344, 340/343, 341/342 (333/350 = + 17) (17) 699 = 341/344, 342/343 (341/344 = + 3) (3) 701 = 343/344 703 = 335/354, 336/353, 337/352, 338/351, 339/350, 340/349, 341/348, 342/347, 343/346, 344/345 (335/354 = + 19) (19) 705 = 344/347, 345/346 (344/347 = + 3) (3) 707 = 343/350, 344/349, 345/348, 346/347 (343/350 = + 7) (7) 709 = 347/348 711 = 347/350, 348/349 (347/350 = + 3) (3) 713 = 338/361, 339/360, 340/359, 341/358, 342/357, 343/356, 344/355, 345/354, 346/353, 347/352, 348/351, 349/350 (338/361 = + 23) (23) 715 = 348/353, 349/352, 350/351 (338/343 = + 5) (5) 717 = 350/353, 351/352 (350/353 = + 3) (3) 719 = 352/353 721 = 350/357, 351/356, 352/355, 353/354 (350/357 = + 7) (7) 723 = 353/356, 354/355 (353/356 = + 3) (3) 725 = 353/358, 354/357, 355/356 (353/358 = + 5) (5) 727 = 356/357 729 = 356/359, 357/358 (356/359 = + 3) (3)18

731 = 350/367, 351/366, 352/365, 353/364, 354/363, 355/362, 356/361, 357/360, 358/359 (350/367 = + 17) (17) 733 = 359/360 735 = 358/363, 359/362, 360/361 (358/363 = + 5) (5) 737 = 356/367, 357/366, 358/365, 359/364, 360/363, 361/362 (312/323 = + 11) (11) 739 = 362/363 741 = 362/365, 363/364 (362/365 = + 3) (3) 743 = 364/365 745 = 363/368, 364/367, 365/366 (363/368 = + 5) (5) 747 = 365/368, 366/367 (365/368 = + 3) (3) 749 = 364/371, 365/370, 366/369, 367/368 (364/371 = + 7) (7) 751 = 368/369 753 = 368/371, 369/370 (368/371 = + 3) (3) 755 = 368/373, 369/372, 370/371 (368/373 = + 5) (5) 757 = 371/372 759 = 371/374, 372/373 (371/374 = + 3) (3) 761 = 373/374 763 = 371/378, 372/377, 373/376, 374/375 (371/378 = + 7) (7) 765 = 374/377, 375/376 (374/377 = + 3) (3) 767 = 370/383, 371/382, 372/381, 373/380, 374/379, 375/378, 376/377 (370/383 = + 13) (13) 769 = 377/378 771 = 377/380, 378/379 (377/380 = + 3) (3) 773 = 379/38019

775 = 378/383, 379/382, 380/381 (378/383 = + 5) (5) 777 = 380/383, 381/382 (380/383 = + 3) (3) 779 = 373/392, 374/391, 375/390, 376/389, 377/388, 378/387, 379/386, 380/385, 381/384, 382/383 (373/392 = + 19) (19) 781 = 378/389, 379/388, 380/387, 381/386, 382/385, 383/384 (378/389 = + 11) (11) 783 = 383/386, 384/385 (383/386 = + 3) (3) 785 = 383/388, 384/387, 385/386 (383/388 = + 5) (5) 787 = 386/387 789 = 386/389, 387/388 (386/389 = + 3) (3) 791 = 385/392, 386/391, 387/390, 388/389 (385/392 = + 7) (7) 793 = 383/396, 384/395, 385/394, 386/393, 387/392, 388/391, 389/390 (383/396 = + 13) (13) 795 = 389/392, 390/391 (389/392 = + 3) (3) 797 = 391/392 799 = 384/401, 385/400, 386/399, 387/398, 388/397, 389/396, 390/395, 391/394, 392/393 (384/401 = + 17) (17) 801 = 392/395, 393/394 (392/395 = + 3) (3) 803 = 389/400, 390/399, 391/398, 392/397, 393/396, 394/395 (389/400 = + 11) (11) 805 = 393/398, 394/397, 395/396 (393/398 = + 5) (5) 807 = 395/398, 396/397 (395/398 = + 3) (3) 809 = 397/398 811 = 398/399 813 = 398/401, 399/400 (398/401 = + 3) (3) 815 = 398/403, 399/402, 400/401 (398/403 = + 5) (5)20

817 = 392/411, 393/410, 394/409, 395/408, 396/407, 397/406, 398/405, 399/404, 400/403, 401/402 (392/411 = + 19) (19) 819 = 401/404, 402/403 (401/404 = + 3) (3) 821 = 403/404 823 = 404/405 825 = 404/407, 405/406 (404/407 = + 3) (3) 827 = 406/407 829 = 407/408 831 = 407/410, 408/409 (407/410 = + 3) (3) 833 = 406/413, 407/412, 408/411, 409/410 (406/413 = + 7) (7) 835 = 408/413, 409/412, 410/411 (408/413 = + 5) (5) 837 = 410/413, 411/412 (410/413 = + 3) (3) 839 = 412/413 841 = 399/428, 400/427, 401/426, 402/425, 403/424, 404/423, 405/422, 406/421, 407/420, 408/419, 409/418, 410/417, 411/416, 412/415, 413/414 (399/428 = + 29) (29) 843 = 413/416, 414/415 (413/416 = + 3) (3) 845 = 413/418, 414/417, 415/416 (413/418 = + 5) (5) 847 = 413/420, 414/419, 415/418, 416/417 (413/420 = + 7) (7) 849 = 416/419, 417/418 (416/419 = + 3) (3) 851 = 407/430, 408/429, 409/428, 410/427, 411/426, 412/425, 413/424, 414/423, 415/422, 416/421, 417/420, 418/419 (407/430 = + 23) (23) 853 = 419/420 855 = 419/422, 420/421 (419/422 = + 3) (3) 857 = 421/422 859 = 422/42321

861 = 422/425, 423/424 (422/425 = + 3) (3) 863 = 424/425 865 = 423/428, 424/427, 425/426 (423/428 = + 5) (5) 867 = 425/428, 426/427 (425/428 = + 3) (3) 869 = 422/433, 423/432, 424/431, 425/430, 426/429, 427/428 (422/433 = + 11) (11) 871 = 422/435, 423/434, 424/433, 425/432, 426/431, 427/430, 428/429 (422/435 = + 13) (13) 873 = 428/431, 429/430 (428/431 = + 3) (3) 875 = 428/433, 429/432, 430/431 (428/433 = + 5) (5) 877 = 431/432 879 = 431/434, 432/433 (431/434 = + 3) (3) 881 = 433/434 883 = 434/435 885 = 434/437, 435/436 (434/437 = + 3) (3) 887 = 436/437 889 = 434/441, 435/440, 436/439, 437/438 (434/441 = + 7) (7) 891 = 437/440, 438/439 (437/440 = + 3) (3) 893 = 430/449, 431/448, 432/447, 433/446, 434/445, 435/444, 436/443, 437/442, 438/441, 439/440 (430/449 = + 19) (19) 895 = 438/443, 439/442, 440/441 (438/443 = + 5) (5) 897 = 440/443, 441/442 (440/443 = + 3) (3) 899 = 428/457, 429/456, 430/455, 431/454, 432/453, 433/452, 434/451, 435/450, 436/449, 437/448, 438/447, 439/446, 440/445, 441/444, 442/443 (428/457 = + 29) (29) 901 = 435/452, 436/451, 437/450, 438/449, 439/448, 440/447, 441/446, 442/445, 443/444 (435/452 = + 17) (17)22

903 = 443/446, 444/445 (443/446 = + 3) (3) 905 = 443/448, 444/447, 445/446 (443/448 = + 5) (5) 907 = 446/447 909 = 446/449, 447/448 (446/449 = + 3) (3) 911 = 448/449 913 = 444/455, 445/454, 446/453, 447/452, 448/451, 449/450 (444/455 = + 11) (11) 915 = 449/452, 450/451 (449/452 = + 3) (3) 917 = 448/455, 449/454, 450/453, 451/452 (448/455 = + 7) (7) 919 = 452/453 921 = 452/453, 453/454 (452/453 = + 3) (3) 923 = 449/461, 449/460, 450/459, 451/458, 452/457, 453/456, 454/455 (449/461 = + 13) (13) 925 = 453/458, 454/457, 455/456 (453/458 = + 5) (5) 927 = 455/458, 456/457 (455/458 = + 3) (3) 929 = 457/458 931 = 455/462, 456/461, 457/460, 458/459 (455/462 = + 7) (7) 933 = 458/461, 459/460 (458/461 = + 3) (3) 935 = 458/463, 459/462, 460/461 (458/463 = + 5) (5) 937 = 461/462 939 = 461/464, 462/463 (461/464 = + 3) (3) 941 = 463/464 943 = 453/476, 454/475, 455/474, 456/473, 457/472, 458/471, 459/470, 460/469, 461/468, 462/467, 463/466, 464/465 (453/476 = + 23) (23) 945 = 464/467, 465/466 (464/467 = + 3) (3)23

947 = 466/467 949 = 461/474, 462/473, 463/472, 464/471, 465/470, 466/469, 467/468 (449/461 = + 13) (13) 951 = 467/470, 468/469 (467/470 = + 3) (3) 953 = 469/470 955 = 468/473, 469/472, 470/471 (468/473 = + 5) (5) 957 = 470/473, 471/472 (470/473 = + 3) (3) 959 = 469/476, 470/475, 471/474, 472/473 (469/476 = + 7) (7) 961 = 458/489, 459/488, 460/487, 461/486, 462/485, 463/484, 464/483, 465/482, 466/481, 467/480, 468/479, 469/478, 470/477, 471/476, 472/475, 473/474 (458/489 = + 31) (31) 963 = 473/476, 474/475 (473/476 = + 3) (3) 965 = 473/478, 474/477, 475/476 (473/478 = + 5) (5) 967 = 476/477 969 = 476/479, 477/478 (476/479 = + 3) (3) 971 = 478/479 973 = 476/483, 477/482, 478/481, 479/480 (476/483 = + 7) (7) 975 = 479/482, 480/481 (479/482 = + 3) (3) 977 = 481/482 979 = 477/488, 478/487, 479/486, 480/485, 481/484, 482/483 (477/488 = + 11) (11) 981 = 482/485, 483/484 (482/485 = + 3) (3) 983 = 484/485 985 = 483/488, 484/487, 485/486 (483/488 = + 5) (5) 987 = 485/488, 486/487 (485/488 = + 3) (3)

24

989 = 476/499, 477/498, 478/497, 479/496, 480/495, 481/494, 482/493, 483/492, 484/491, 485/490, 486/489, 487/488 (476/499 = + 23) (23) 991 = 488/489 993 = 488/491, 489/490 (488/491 = + 3) (3) 995 = 488/493, 489/492, 490/491 (488/493 = + 5) (5) 997 = 491/492 999 = 491/494, 492/493 (491/494 = + 3) (3)

7132283382211753065664038103921 = 3566141691105876532832019051960/3566141691105876532832019051961

118558347188026655500106547231096910504441858017 = 59279173594013327750053273615548455252220929007/592791735940133277 50053273615548455252220929010, 59279173594013327750053273615548455252220929008/ 59279173594013327750053273615548455252220929009 (59279173594013327750053273615548455252220929007/ 59279173594013327750053273615548455252220929010 = + 3) (3)

24741128862775614761078532487282208383185206912457942962646484375 = 12370564431387807380539266243641104191592603456228971481323242185/ 12370564431387807380539266243641104191592603456228971481323242190, 12370564431387807380539266243641104191592603456228971481323242186/ 12370564431387807380539266243641104191592603456228971481323242189, 12370564431387807380539266243641104191592603456228971481323242187/25

12370564431387807380539266243641104191592603456228971481323242188 (12370564431387807380539266243641104191592603456228971481323242185/ 12370564431387807380539266243641104191592603456228971481323242190 = + 5) (5)

185323955009471744507093833849366798683834244443114056794632807824 05796233163977 = 926619775047358722535469169246833993419171222215570283973164039120 2898116581988/9266197750473587225354691692468339934191712222155702 839731640391202898116581989

207472224677348520782169522210760858748099647472111729275299258991 2196684750549658310084416732550077 = 103736112338674260391084761105380429374049823736055864637649629495 6098342375274829155042208366275038/1037361123386742603910847611053 804293740498237360558646376496294956098342375274829155042208366275 039

418694384508150135841573026266725373160871802417248330252097316337 271746706959547700749430290014885946761737884442289409357 = 209347192254075067920786513133362686580435901208624165126048658168 635873353479773850374715145007442973380868942221144704675/ 209347192254075067920786513133362686580435901208624165126048658168 635873353479773850374715145007442973380868942221144704682, 209347192254075067920786513133362686580435901208624165126048658168 635873353479773850374715145007442973380868942221144704676/ 209347192254075067920786513133362686580435901208624165126048658168 635873353479773850374715145007442973380868942221144704681,26

209347192254075067920786513133362686580435901208624165126048658168 635873353479773850374715145007442973380868942221144704677/ 209347192254075067920786513133362686580435901208624165126048658168 635873353479773850374715145007442973380868942221144704680, 209347192254075067920786513133362686580435901208624165126048658168 635873353479773850374715145007442973380868942221144704678/ 209347192254075067920786513133362686580435901208624165126048658168 635873353479773850374715145007442973380868942221144704679 (20934719225407506792078651313336268658043590120862416512604865816 8635873353479773850374715145007442973380868942221144704675/ 209347192254075067920786513133362686580435901208624165126048658168 635873353479773850374715145007442973380868942221144704682 = + 7) (7)

Sergio Aschero (Doctor en Musicologa) [email protected] Avenida Roca 1669 (1870) Avellaneda / Provincia de Buenos Aires / Argentina http://ascheropus.blogspot.com

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