Indice
- 1. Pagina principale
- 2. Congettura sui numeri primi (I)
- 3. Congettura sui numeri primi (II)
- 4. Congettura sui numeri super iperperfetti
- 5. Congettura sui primi geometricamente deboli
- 6. Congetture sulla rappresentazione dei naturali tramite numeri pentagonali generalizzati
- 7. Congetture sulla rappresentazione dei naturali tramite numeri piramidali
- 8. Congetture sulla rappresentazione dei naturali tramite numeri piramidali centrati
- 9. In costruzione
- 10. In costruzione
- 11. Congettura sulla rappresentazione dei numeri pari come somma di quadrati e primi
- 12. Congettura su serie convergenti a potenze di π
- 13. In costruzione
- 14. Congetture sulla rappresentazione dei naturali come somme di numeri poligonali generalizzati
Per quanto riguarda la rappresentazione dei numeri naturali tramite numeri pentagonali e numeri pentagonali generalizzati, il primo gruppo di congetture afferma che ogni numero naturale si possa rappresentare tramite una delle seguenti somme:
P”x + P”y + Pz;
P”x + Py + 2Pz;
P”x + Py + 3P”z;
P”x + P”y + 3Pz;
P”x + Py + 4P”z;
P”x + P”y + 4Pz;
P”x + P”y + 5Pz;
P”x + P”y + 6Pz;
P”x + P”y + 8Pz;
P”x + P”y + 10Pz;
P”x + 2P”y + 2Pz;
P”x + 2P”y + 2P”z;
Px + 2P”y + 3P”z;
P”x + 2Py + 3P”z;
P”x + 2Py + 4P”z;
P”x + 3Py + 4P”z;
P”x + 3P”y + 4Pz;
P”x + 3Py + 6P”z.
Il secondo gruppo di congetture afferma invece che le combinazioni riportate nella tabella seguente possono rappresentare tutti i numeri naturali, tranne un numero finito di eccezioni.
|
Somma |
Numero di eccezioni |
Eccezioni |
|
P”x + Py + Pz |
1 |
33 |
|
Px + Py + 2Pz |
49 |
9, 18, 21, 28, 31, 39, 43, 55, 69, 74, 89, 90, 98, 109, 111, 113, 134, 135, 144, 193, 202, 214, 230, 243, 260, 265, 273, 275, 310, 510, 553, 698, 699, 749, 773, 780, 865, 878, 945, 965, 1219, 1398, 1413, 2153, 2168, 2335, 2828, 3178, 3793 |
|
Px + Py + 2P”z |
5 |
18, 98, 113, 193, 553 |
|
Px + Py + 3Pz |
85 |
7, 11, 14, 18, 19, 29, 31, 33, 45, 61, 64, 65, 69, 77, 81, 84, 91, 94, 98, 103, 116, 126, 131, 134, 135, 147, 151, 164, 169, 173, 174, 185, 194, 208, 221, 229, 231, 236, 238, 241, 256, 279, 294, 306, 319, 343, 344, 409, 410, 439, 467, 507, 546, 571, 634, 684, 714, 795, 824, 831, 854, 915, 971, 999, 1054, 1144, 1193, 1260, 1266, 1300, 1306, 1323, 1589, 1624, 1635, 1902, 2034, 2161, 2236, 2541, 2881, 3119, 3259, 6006, 6471 |
|
Px + Py + 3P”z |
3 |
14, 64, 169 |
|
P”x + Py + 3Pz |
2 |
33, 208 |
|
Px + Py + 4Pz |
105 |
3, 7, 8, 11, 15, 18, 19, 29, 41, 45, 46, 59, 62, 66, 68, 69, 78, 80, 81, 85, 87, 103, 107, 116, 120, 136, 148, 155, 173, 178, 179, 183, 186, 189, 195, 199, 212, 220, 223, 243, 253, 271, 278, 305, 311, 323, 345, 348, 360, 363, 389, 393, 416, 417, 421, 435, 444, 455, 458, 487, 500, 522, 529, 575, 584, 647, 654, 675, 688, 696, 718, 753, 780, 851, 858, 859, 869, 917, 955, 971, 1053, 1125, 1191, 1208, 1214, 1341, 1403, 1421, 1598, 1628, 1663, 1686, 1790, 1905, 1940, 2563, 2824, 2963, 3293, 3348, 3473, 3536, 7110, 12869, 16892 |
|
Px + Py + 4P”z |
13 |
3, 7, 11, 15, 19, 46, 69, 136, 220, 311, 323, 500, 718 |
|
P”x + Py + 4Pz |
4 |
59, 80, 87, 654 |
|
Px + Py + 5Pz |
172 |
3, 4, 8, 9, 14, 16, 19, 20, 21, 33, 43, 46, 50, 53, 54, 55, 58, 64, 67, 74, 79, 85, 89, 90, 99, 101, 103, 106, 108, 113, 124, 125, 128, 136, 138, 141, 149, 156, 158, 159, 160, 163, 169, 179, 190, 191, 194, 195, 204, 208, 218, 225, 229, 230, 233, 243, 254, 263, 276, 281, 283, 300, 301, 310, 316, 333, 345, 349, 354, 361, 366, 368, 375, 383, 396, 410, 414, 418, 433, 438, 456, 459, 469, 509, 515, 525, 529, 541, 550, 568, 571, 580, 613, 633, 639, 645, 710, 739, 756, 774, 778, 786, 814, 820, 823, 840, 851, 859, 863, 888, 900, 946, 948, 956, 978, 1016, 1044, 1103, 1134, 1194, 1229, 1263, 1289, 1393, 1406, 1425, 1464, 1495, 1535, 1579, 1626, 1705, 1745, 1786, 1803, 1863, 1874, 1933, 1939, 1946, 2049, 2069, 2116, 2206, 2226, 2286, 2330, 2339, 2479, 2484, 2639, 2809, 2850, 2890, 2935, 3236, 3319, 3561, 4048, 4244, 4246, 4643, 4705, 4734, 4890, 4958, 5433, 5891, 6351, 7349, 7588, 11835 |
|
Px + Py + 5P”z |
38 |
3, 4, 8, 9, 14, 19, 21, 43, 53, 55, 64, 74, 89, 90, 101, 113, 125, 141, 158, 159, 163, 169, 195, 229, 230, 254, 301, 316, 354, 361, 383, 438, 515, 613, 739, 1464, 2069, 2850 |
|
P”x + Py + 5Pz |
3 |
4, 9, 64 |
|
P”x + Py + 5P”z |
2 |
4, 9 |
|
Px + Py + 6Pz |
133 |
3, 4, 9, 14, 15, 20, 21, 25, 26, 37, 38, 39, 45, 48, 49, 55, 59, 60, 61, 67, 68, 80, 83, 90, 91, 109, 113, 115, 125, 126, 130, 131, 136, 141, 153, 160, 161, 171, 178, 185, 191, 200, 208, 213, 219, 230, 231, 235, 242, 255, 263, 279, 295, 297, 301, 325, 326, 373, 375, 396, 405, 413, 415, 439, 440, 461, 465, 480, 485, 510, 514, 526, 556, 581, 585, 611, 643, 650, 675, 680, 711, 760, 771, 776, 790, 828, 865, 881, 885, 935, 942, 973, 986, 1005, 1065, 1120, 1151, 1181, 1245, 1310, 1425, 1536, 1810, 1816, 1973, 2006, 2075, 2078, 2091, 2306, 2355, 2435, 2476, 2528, 2661, 2675, 2806, 2845, 2895, 2901, 3146, 3536, 3560, 3600, 3706, 4335, 4468, 5105, 5190, 5781, 8963, 11416, 18395 |
|
Px + Py + 6P”z |
23 |
3, 4, 9, 15, 20, 21, 26, 37, 38, 45, 49, 60, 61, 67, 80, 131, 136, 255, 325, 375 396, 461, 711 |
|
P”x + Py + 6Pz |
8 |
4, 39, 55, 60, 90, 136, 235, 1151 |
|
P”x + Py + 6P”z |
1 |
4 |
|
Px + Py + 8Pz |
168 |
3, 4, 7, 11, 15, 16, 19, 26, 28, 29, 33, 37, 38, 39, 49, 54, 58, 61, 66, 68, 69, 72, 77, 85, 88, 89, 95, 99, 107, 116, 124, 128, 134, 138, 141, 149, 155, 156, 163, 164, 172, 173, 174, 191, 194, 205, 226, 229, 231, 243, 244, 257, 265, 266, 271, 275, 279, 289, 291, 312, 313, 334, 340, 345, 354, 359, 368, 369, 380, 390, 393, 399, 415, 429, 436, 439, 441, 449, 450, 480, 492, 504, 509, 524, 538, 563, 581, 605, 621, 627, 647, 683, 711, 733, 734, 754, 781, 810, 826, 843, 856, 880, 889, 894, 909, 979, 1012, 1019, 1039, 1047, 1051, 1073, 1083, 1108, 1149, 1153, 1164, 1169, 1239, 1244, 1250, 1318, 1320, 1323, 1391, 1416, 1556, 1684, 1804, 1824, 1836, 1843, 1922, 1957, 1980, 2096, 2109, 2154, 2165, 2280, 2287, 2551, 2644, 2669, 2784, 3120, 3166, 3179, 3294, 3344, 3498, 3572, 3926, 4235, 4516, 4534, 4589, 4638, 5080, 5449, 5884, 5983, 6802, 7655, 8304, 8605, 9491, 20704 |
|
Px + Py + 8P”z |
38 |
3, 4, 7, 11, 15, 19, 37, 49, 54, 77, 85, 88, 95, 99, 116, 124, 128, 141, 163, 194, 205, 229, 257, 289, 291, 334, 340, 345, 359, 369, 399, 415, 605, 627, 781, 979, 1169, 1556 |
|
P”x + Py + 8Pz |
6 |
4, 33, 68, 95, 291, 1391 |
|
P”x + Py + 8P”z |
1 |
4 |
|
Px + Py + 9Pz |
215 |
3, 4, 7, 8, 16, 18, 20, 25, 28, 29, 30, 37, 38, 39, 41, 42, 48, 54, 59, 64, 74, 76, 77, 78, 83, 87, 88, 90, 94, 98, 99, 100, 103, 107, 112, 119, 124, 128, 133, 134, 141, 151, 153, 156, 158, 169, 170, 173, 175, 182, 192, 206, 214, 216, 217, 223, 228, 239, 240, 244, 263, 264, 266, 274, 281, 283, 286, 294, 305, 310, 324, 329, 334, 346, 358, 363, 368, 389, 391, 393, 399, 403, 405, 414, 415, 416, 418, 428, 440, 448, 451, 452, 453, 462, 474, 479, 501, 509, 514, 518, 548, 558, 559, 572, 593, 606, 638, 659, 671, 683, 684, 688, 689, 690, 714, 754, 756, 763, 774, 802, 825, 831, 844, 856, 865, 888, 894, 924, 959, 979, 984, 986, 1000, 1019, 1020, 1044, 1086, 1095, 1113, 1133, 1143, 1154, 1169, 1187, 1294, 1295, 1308, 1353, 1369, 1371, 1399, 1404, 1420, 1459, 1508, 1509, 1519, 1572, 1664, 1683, 1708, 1712, 1791, 1844, 1881, 1999, 2064, 2338, 2428, 2429, 2449, 2453, 2527, 2548, 2661, 2668, 2674, 2703, 2713, 2804, 2849, 2854, 2967, 3038, 3108, 3473, 3497, 3535, 3639, 3648, 3948, 4114, 4243, 4264, 4374, 4584, 4605, 4649, 4714, 4959, 5103, 5128, 5159, 5190, 5699, 6369, 6993, 7204, 7321, 8029, 8351, 8488, 11778, 12228, 12333 |
|
Px + Py + 9P”z |
38 |
3, 4, 7, 8, 16, 25, 29, 37, 38, 39, 48, 59, 77, 78, 83, 94, 112, 124, 128, 151, 153, 173, 223, 283, 294, 329, 334, 358, 368, 389, 501, 559, 593, 683, 684, 754, 802, 979 |
|
P”x + Py + 9Pz |
7 |
4, 18, 30, 39, 175, 182, 399 |
|
P”x + Py + 9P”z |
2 |
4, 39 |
|
P”x + P”y + 9Pz |
1 |
182 |
|
Px + Py + 10Pz |
355 |
3, 4, 7, 8, 9, 14, 18, 19, 21, 25, 26, 28, 29, 30, 31, 38, 39, 41, 42, 43, 48, 49, 53, 58, 59, 64, 65, 68, 69, 76, 78, 79, 87, 88, 89, 91, 95, 98, 99, 100, 108, 109, 110, 111, 116, 119, 134, 135, 138, 141, 148, 151, 158, 159, 161, 163, 165, 166, 169, 170, 173, 174, 175, 182, 185, 192, 199, 201, 203, 204, 205, 214, 216, 223, 228, 229, 235, 236, 239, 240, 250, 251, 253, 273, 274, 275, 281, 285, 286, 289, 294, 305, 310, 314, 315, 320, 323, 326, 328, 333, 336, 344, 346, 353, 358, 361, 369, 378, 383, 393, 395, 399, 403, 409, 417, 424, 428, 434, 439, 444, 445, 449, 453, 469, 474, 483, 491, 498, 514, 535, 536, 553, 563, 568, 576, 589, 610, 616, 621, 629, 658, 665, 668, 669, 676, 679, 684, 693, 704, 728, 733, 739, 746, 749, 759, 781, 784, 788, 791, 799, 820, 823, 824, 830, 834, 838, 858, 859, 866, 879, 885, 895, 923, 929, 931, 950, 965, 989, 996, 1000, 1048, 1061, 1068, 1075, 1083, 1106, 1110, 1113, 1124, 1136, 1153, 1155, 1198, 1209, 1220, 1236, 1239, 1246, 1268, 1271, 1273, 1281, 1328, 1330, 1334, 1344, 1369, 1391, 1393, 1411, 1428, 1444, 1446, 1493, 1495, 1526, 1540, 1541, 1566, 1585, 1594, 1610, 1613, 1615, 1624, 1642, 1671, 1674, 1708, 1733, 1750, 1786, 1866, 1869, 1879, 1909, 1923, 1930, 1960, 1963, 1985, 2009, 2049, 2056, 2076, 2088, 2111, 2119, 2153, 2154, 2188, 2195, 2231, 2265, 2321, 2326, 2338, 2371, 2405, 2489, 2490, 2518, 2583, 2611, 2639, 2653, 2671, 2684, 2700, 2826, 2839, 2903, 2958, 2966, 2986, 3000, 3078, 3195, 3203, 3224, 3230, 3385, 3458, 3459, 3475, 3570, 3651, 3745, 3759, 3779, 3781, 3838, 3965, 4036, 4078, 4165, 4239, 4246, 4324, 4544, 4701, 4753, 4764, 4785, 4893, 4924, 4926, 4953, 4961, 5080, 5086, 5128, 5213, 5239, 5323, 5366, 5605, 5645, 5824, 5939, 6118, 6578, 6741, 6794, 6795, 6836, 7560, 7578, 7600, 7761, 8103, 8276, 8748, 9238, 9265, 9730, 9769, 10665, 11373, 11571, 11791, 13108, 13528, 14623, 15505, 16076, 16231, 17560, 17896, 18680, 19891, 21949, 36694, 40915, 43705 |
|
Px + Py + 10P”z |
72 |
3, 4, 7, 8, 9, 14, 18, 19, 28, 29, 31, 38, 39, 41, 48, 49, 53, 58, 59, 65, 68, 69, 78, 79, 88, 89, 98, 99, 100, 108, 109, 111, 116, 119, 135, 148, 158, 161, 169, 203, 205, 214, 228, 239, 240, 253, 274, 275, 286, 305, 314, 320, 328, 336, 344, 383, 393, 439, 453, 491, 535, 616, 629, 679, 859, 965, 1136, 1239, 1271, 1328, 1674, 2490 |
|
P”x + Py + 10Pz |
14 |
4, 9, 21, 28, 43, 49, 59, 163, 175, 203, 229, 274, 285, 669 |
|
P”x + Py + 10P”z |
3 |
4, 9, 59 |
|
P”x + P”y + 10Pz |
3 |
21, 49, 163 |
|
Px + 2Py + 2Pz |
119 |
6, 8, 18, 19, 23, 28, 30, 33, 40, 41, 43, 50, 52, 57, 62, 64, 65, 67, 78, 86, 87, 91, 98, 101, 108, 111, 120, 122, 125, 128, 130, 132, 133, 135, 144, 156, 159, 166, 167, 170, 183, 190, 192, 203, 227, 252, 253, 260, 262, 265, 268, 272, 275, 284, 294, 310, 323, 338, 344, 347, 383, 415, 436, 439, 450, 458, 461, 463, 467, 482, 498, 517, 520, 554, 593, 648, 657, 673, 686, 708, 720, 747, 775, 778, 877, 902, 918, 933, 970, 1004, 1098, 1122, 1163, 1238, 1254, 1263, 1402, 1408, 1490, 1560, 1594, 1647, 1690, 1752, 1793, 1883, 2117, 2298, 2309, 2398, 2652, 2753, 2782, 3354, 4363, 4926, 5538, 5678, 7727 |
|
Px + 2P”y + 2Pz |
3 |
8, 23, 78 |
|
P”x + 2Py + 2Pz |
17 |
8, 18, 23, 43, 62, 65, 78, 122, 132, 135, 156, 192, 253, 323, 463, 482, 708 |
|
Px + 2Py + 3Pz |
93 |
9, 19, 21, 23, 31, 33, 34, 42, 55, 57, 63, 84, 93, 96, 99, 100, 104, 113, 123, 126, 133, 135, 159, 174, 195, 216, 231, 240, 243, 265, 273, 303, 309, 318, 336, 384, 405, 408, 411, 441, 489, 518, 546, 555, 558, 569, 618, 657, 671, 693, 699, 723, 798, 801, 831, 834, 909, 924, 939, 996, 1113, 1155, 1254, 1263, 1365, 1386, 1449, 1689, 1725, 1750, 1848, 1899, 2085, 2088, 2228, 2295, 2319, 2376, 2970, 2973, 2982, 3174, 3954, 4398, 4587, 5049, 5259, 6348, 6729, 9603, 10899, 16578, 20988 |
|
Px + 2Py + 3P”z |
5 |
19, 34, 63, 84, 99 |
|
Px + 2P”y + 3Pz |
11 |
21, 23, 63, 104, 113, 133, 309, 518, 569, 671, 2228 |
|
P”x + 2Py + 3Pz |
7 |
21, 23, 33, 63, 135, 309, 558 |
|
P”x + 2Py + 3P”z |
2 |
23, 63 |
|
Px + 2Py + 4Pz |
56 |
8, 13, 17, 38, 43, 47, 54, 67, 68, 78, 87, 108, 126, 129, 135, 138, 146, 148, 171, 174, 197, 208, 213, 221, 227, 231, 262, 313, 314, 347, 348, 363, 367, 381, 415, 454, 496, 502, 511, 601, 628, 642, 649, 689, 887, 1004, 1195, 1217, 1258, 1336, 1733, 1874, 2541, 2602, 3447, 4451 |
|
Px + 2Py + 4P”z |
7 |
17, 47, 68, 138, 146, 363, 496 |
|
Px + 2P”y + 4Pz |
6 |
17, 38, 47, 54, 108, 138 |
|
P”x + 2Py + 4Pz |
3 |
38, 47, 78 |
|
Px + 2P”y + 4P”z |
1 |
138 |
|
P”x + 2P”y + 4Pz |
2 |
38, 47 |
|
Px + 2Py + 6Pz |
118 |
4, 19, 23, 26, 27, 34, 39, 47, 48, 58, 60, 63, 64, 68, 69, 90, 93, 99, 106, 115, 126, 129, 148, 150, 160, 163, 165, 173, 180, 201, 223, 228, 229, 231, 243, 267, 270, 274, 283, 288, 298, 300, 314, 333, 339, 345, 368, 375, 391, 405, 414, 415, 418, 439, 460, 480, 498, 513, 612, 621, 643, 720, 729, 735, 740, 778, 795, 879, 911, 918, 943, 970, 1044, 1113, 1125, 1149, 1153, 1251, 1299, 1315, 1328, 1350, 1380, 1464, 1653, 1672, 1674, 1752, 1803, 1845, 1945, 2049, 2070, 2088, 2538, 2694, 2961, 3168, 3375, 3600, 3868, 3963, 4185, 4239, 4248, 4689, 4863, 5328, 6099, 7344, 7593, 8118, 8208, 8973, 11583, 14628, 19290, 26073 |
|
Px + 2Py + 6P”z |
10 |
4, 39, 60, 69, 99, 150, 223, 267, 480, 1380 |
|
Px + 2P”y + 6Pz |
15 |
23, 27, 47, 68, 99, 173, 223, 243, 288, 314, 368, 414, 418, 498, 1328 |
|
P”x + 2Py + 6Pz |
16 |
19, 27, 60, 68, 99, 201, 243, 288, 405, 414, 418, 498, 720, 740, 1672, 1845 |
|
Px + 2P”y + 6P”z |
1 |
223 |
|
P”x + 2Py + 6P”z |
1 |
60 |
|
P”x + 2P”y + 6Pz |
3 |
68, 99, 418 |
|
Px + 2Py + 8Pz |
118 |
4, 6, 16, 17, 21, 26, 27, 28, 31, 34, 38, 39, 48, 58, 60, 63, 68, 73, 81, 104, 109, 123, 126, 130, 138, 139, 146, 150, 151, 158, 165, 173, 174, 182, 207, 230, 238, 244, 245, 253, 261, 263, 266, 267, 273, 307, 314, 322, 347, 373, 377, 405, 406, 424, 438, 451, 468, 491, 504, 543, 557, 559, 578, 588, 648, 685, 689, 694, 714, 742, 769, 783, 798, 819, 823, 865, 978, 986, 988, 1049, 1070, 1113, 1140, 1154, 1173, 1198, 1218, 1294, 1295, 1308, 1320, 1369, 1376, 1388, 1446, 1516, 1578, 1673, 1714, 1847, 1868, 1882, 1896, 1923, 1943, 2198, 2426, 2548, 2618, 2673, 2806, 3353, 3773, 3780, 4220, 5716, 6391, 19863 |
|
Px + 2Py + 8P”z |
11 |
4, 6, 34, 39, 73, 104, 139, 146, 174, 182, 238 |
|
Px + 2P”y + 8Pz |
17 |
6, 21, 28, 48, 58, 68, 109, 182, 238, 244, 266, 273, 451, 865,1218, 1294, 1516 |
|
P”x + 2Py + 8Pz |
20 |
6, 16, 21, 27, 38, 60, 63, 73, 123, 130, 139, 151, 245, 267, 273, 405, 557, 1049, 1218, 2548 |
|
Px + 2P”y + 8P”z |
1 |
6 |
|
P”x + 2Py + 8P”z |
2 |
6, 139 |
|
P”x + 2P”y + 8Pz |
1 |
273 |
|
Px + 3Py + 3Pz |
318 |
2, 9, 10, 13, 14, 17, 21, 24, 26, 29, 32, 33, 34, 43, 45, 46, 47, 49, 55, 59, 60, 62, 64, 68, 75, 79, 80, 83, 89, 96, 97, 99, 111, 112, 115, 118, 119, 126, 129, 134, 138, 149, 150, 152, 162, 166, 170, 174, 177, 185, 187, 195, 199, 205, 208, 209, 221, 229, 234, 236, 239, 243, 244, 252, 255, 256, 266, 267, 269, 271, 272, 274, 275, 294, 299, 300, 304, 310, 321, 322, 325, 331, 335, 339, 340, 362, 372, 374, 380, 384, 390, 397, 410, 413, 426, 437, 444, 449, 453, 454, 463, 465, 469, 474, 475, 503, 510, 514, 515, 524, 551, 554, 560, 570, 572, 577, 584, 592, 607, 617, 619, 624, 625, 643, 661, 663, 670, 675, 679, 689, 694, 699, 713, 726, 729, 734, 752, 755, 769, 802, 810, 824, 825, 849, 850, 859, 885, 893, 908, 936, 947, 969, 974, 979, 999, 1000, 1034, 1035, 1047, 1053, 1059, 1079, 1090, 1099, 1105, 1112, 1115, 1125, 1147, 1156, 1175, 1189, 1196, 1197, 1225, 1230, 1242, 1244, 1249, 1285, 1314, 1324, 1337, 1399, 1419, 1449, 1452, 1464, 1475, 1508, 1510, 1511, 1539, 1649, 1670, 1699, 1704, 1740, 1762, 1781, 1816, 1834, 1844, 1860, 1933, 1938, 2000, 2025, 2044, 2049, 2102, 2105, 2110, 2154, 2204, 2220, 2225, 2284, 2295, 2369, 2379, 2385, 2420, 2442, 2467, 2477, 2515, 2530, 2589, 2617, 2639, 2645, 2652, 2689, 2740, 2749, 2784, 2785, 2822, 2847, 2890, 2938, 3009, 3155, 3160, 3246, 3289, 3300, 3342, 3344, 3478, 3599, 3624, 3625, 3727, 3810, 3874, 3959, 3961, 3990, 4212, 4225, 4387, 4389, 4409, 4472, 4695, 4815, 4989, 5115, 5120, 5200, 5259, 5529, 5549, 5599, 5759, 5795, 5974, 6004, 6044, 6067, 6340, 6579, 7024, 7360, 7439, 7475, 7489, 7615, 7624, 8599, 8755, 9035, 9149, 9215, 9254, 9567, 9735, 10029, 11360, 12314, 12500, 12519, 12610, 12949, 13012, 13599, 14740, 14860, 15084, 15965, 18434, 20004, 20354, 29004, 32014, 36760 |
|
Px + 3P”y + 3Pz |
9 |
2, 13, 17, 32, 34, 55, 68, 89, 97 |
|
P”x + 3Py + 3Pz |
33 |
9, 14, 24, 34, 47, 49, 59, 64, 89, 97, 99, 111, 119, 150, 152, 174, 199, 243, 244, 255, 269, 274, 300, 619, 729, 769, 850, 969, 1189, 1244, 1452, 1510, 2784 |
|
Px + 3P”y + 3P”z |
3 |
2, 55, 68 |
|
P”x + 3P”y + 3Pz |
1 |
34 |
|
Px + 3Py + 4Pz |
104 |
2, 6, 10, 11, 13, 14, 17, 18, 30, 33, 34, 43, 44, 46, 59, 65, 69, 72, 76, 79, 80, 81, 94, 97, 116, 122, 134, 150, 151, 163, 170, 171, 172, 182, 186, 200, 221, 234, 237, 238, 256, 278, 286, 304, 312, 326, 339, 341, 342, 348, 354, 374, 389, 398, 431, 437, 458, 482, 489, 498, 499, 507, 510, 514, 520, 562, 564, 566, 589, 612, 619, 664, 688, 706, 711, 787, 816, 874, 884, 902, 923, 930, 963, 970, 1014, 1035, 1180, 1212, 1348, 1382, 1406, 1411, 1464, 1838, 1938, 2218, 2317, 2334, 2512, 3104, 3114, 4258, 5078, 11554 |
|
Px + 3Py + 4P”z |
19 |
2, 6, 10, 14, 17, 18, 34, 122, 150, 151, 234, 237, 256, 342, 398, 482, 564, 566, 706 |
|
Px + 3P”y + 4Pz |
11 |
2, 13, 14, 17, 34, 44, 97, 122, 150, 200, 234 |
|
P”x + 3Py + 4Pz |
9 |
13, 69, 79, 94, 122, 134, 286, 354, 489 |
|
Px + 3P”y + 4P”z |
2 |
2, 17 |
|
Px + 3Py + 6Pz |
162 |
2, 13, 17, 19, 23, 24, 29, 32, 39, 40, 49, 53, 59, 61, 62, 63, 69, 74, 82, 83, 86, 89, 90, 102, 104, 114, 119, 121, 124, 129, 141, 149, 155, 161, 163, 172, 174, 193, 196, 201, 207, 236, 244, 254, 257, 265, 266, 271, 273, 274, 278, 279, 299, 301, 303, 312, 324, 325, 340, 365, 367, 371, 380, 388, 390, 404, 410, 429, 437, 439, 444, 469, 475, 479, 499, 501, 504, 506, 509, 520, 545, 554, 571, 581, 597, 614, 627, 634, 649, 663, 676, 689, 694, 713, 716, 719, 720, 732, 755, 761, 779, 801, 831, 843, 844, 851, 909, 919, 922, 969, 974, 1027, 1111, 1119, 1175, 1186, 1213, 1219, 1255, 1325, 1329, 1342, 1357, 1374, 1391, 1474, 1489, 1495, 1511, 1516, 1546, 1664, 1675, 1709, 1906, 1970, 2039, 2091, 2099, 2144, 2207, 2244, 2316, 2459, 2581, 2614, 3190, 3276, 3674, 3686, 4199, 4230, 4514, 4836, 5081, 5386, 6466, 7891, 9415, 9489, 14494, 17718 |
|
Px + 3Py + 6P”z |
27 |
2, 19, 23, 29, 40, 59, 61, 74, 86, 89, 114, 121, 124, 155, 201, 236, 279, 324, 475, 506, 597, 614, 634, 1489, 1664, 1675, 3686 |
|
Px + 3P”y + 6Pz |
23 |
2, 19, 23, 29, 40, 53, 59, 61, 69, 74, 114, 124, 141, 149, 174, 207, 279, 410, 581, 779, 2144, 2614, 3686 |
|
P”x + 3Py + 6Pz |
18 |
19, 39, 53, 69, 104, 114, 124, 163, 207, 257, 271, 299, 312, 404, 444, 571, 922, 1219 |
|
Px + 3P”y + 6P”z |
6 |
2, 29, 59, 61, 74, 124 |
|
P”x + 3P”y + 6Pz |
1 |
114 |
|
Px + 3Py + 7Pz |
238 |
2, 6, 9, 13, 14, 17, 18, 21, 24, 26, 28, 30, 31, 33, 46, 49, 52, 53, 56, 59, 63, 64, 68, 69, 75, 79, 81, 82, 90, 91, 97, 98, 103, 115, 116, 118, 126, 129, 131, 133, 137, 144, 146, 149, 164, 168, 173, 177, 178, 184, 187, 196, 199, 203, 206, 207, 209, 228, 231, 233, 234, 235, 236, 243, 251, 256, 266, 273, 278, 279, 289, 291, 292, 300, 301, 304, 317, 321, 324, 328, 338, 339, 341, 343, 344, 347, 349, 378, 389, 406, 418, 422, 438, 446, 448, 451, 453, 466, 472, 473, 479, 501, 508, 514, 549, 551, 553, 555, 571, 573, 583, 584, 588, 592, 609, 623, 629, 632, 639, 653, 675, 676, 693, 696, 699, 755, 787, 790, 794, 812, 821, 829, 864, 878, 879, 898, 913, 916, 956, 993, 1043, 1054, 1059, 1077, 1092, 1119, 1126, 1137, 1178, 1188, 1193, 1199, 1204, 1229, 1256, 1258, 1306, 1336, 1341, 1343, 1384, 1389, 1396, 1403, 1414, 1487, 1533, 1581, 1589, 1609, 1662, 1694, 1714, 1738, 1743, 1851, 1909, 2001, 2018, 2054, 2061, 2113, 2130, 2201, 2205, 2214, 2253, 2263, 2370, 2379, 2389, 2488, 2496, 2579, 2583, 2650, 2769, 2834, 2856, 2866, 2933, 3031, 3219, 3279, 3281, 3358, 3437, 3448, 3483, 3517, 3646, 3758, 3771, 3813, 3904, 4174, 4238, 4469, 4533, 4603, 4724, 5432, 5841, 6253, 6571, 6781, 6864, 6883, 6923, 8101, 9208, 9793, 12028, 13524, 14513, 15603, 25349, 31909 |
|
Px + 3Py + 7P”z |
56 |
2, 6, 9, 13, 21, 24, 28, 31, 33, 46, 56, 59, 63, 75, 79, 82, 91, 98, 118, 126, 129, 133, 149, 164, 173, 184, 196, 199, 231, 234, 235, 243, 251, 266, 273, 278, 279, 324, 349, 389, 446, 448, 479, 508, 514, 571, 583, 623, 794, 821, 956, 1341, 1851, 2130, 2933, 3358 |
|
Px + 3P”y + 7Pz |
23 |
2, 9, 17, 24, 30, 31, 49, 59, 69, 75, 82, 116, 146, 168, 196, 203, 304, 514, 609, 878, 1059, 1204, 1389 |
|
P”x + 3Py + 7Pz |
33 |
6, 13, 21, 28, 31, 46, 49, 56, 59, 63, 82, 98, 118, 137, 149, 206, 233, 251, 273, 328, 349, 378, 389, 422, 453, 473, 501, 623, 916, 956, 1204, 2018, 2054 |
|
Px + 3P”y + 7P”z |
4 |
2, 9, 24, 31 |
|
P”x + 3Py + 7P”z |
9 |
6, 13, 28, 46, 63, 98, 118, 349, 623 |
|
P”x + 3P”y + 7Pz |
2 |
31, 49 |
|
Px + 3Py + 8Pz |
227 |
2, 6, 7, 10, 14, 17, 18, 19, 21, 26, 29, 31, 32, 34, 39, 42, 47, 53, 57, 61, 63, 64, 68, 69, 72, 80, 82, 83, 84, 89, 102, 119, 122, 124, 126, 129, 130, 138, 139, 142, 143, 149, 151, 152, 155, 159, 170, 171, 178, 182, 186, 190, 195, 207, 208, 209, 229, 235, 238, 239, 248, 257, 259, 260, 266, 267, 273, 274, 294, 297, 299, 314, 322, 325, 326, 339, 340, 344, 348, 349, 362, 378, 382, 405, 414, 415, 417, 427, 439, 454, 467, 473, 507, 511, 519, 534, 546, 554, 559, 574, 581, 600, 602, 615, 617, 641, 649, 663, 679, 683, 697, 724, 729, 745, 768, 779, 780, 788, 795, 798, 817, 835, 845, 854, 879, 882, 897, 909, 930, 942, 950, 966, 970, 989, 1011, 1015, 1029, 1048, 1054, 1085, 1105, 1131, 1134, 1144, 1145, 1152, 1235, 1239, 1319, 1354, 1384, 1389, 1391, 1395, 1425, 1484, 1510, 1519, 1559, 1582, 1610, 1635, 1662, 1789, 1872, 1944, 1974, 2114, 2133, 2182, 2239, 2254, 2324, 2385, 2390, 2434, 2469, 2496, 2502, 2549, 2679, 2720, 2749, 2814, 2869, 2879, 2895, 3232, 3449, 3482, 3519, 3534, 3599, 3640, 3692, 4002, 4019, 4120, 4282, 4809, 4939, 5290, 5352, 5477, 5695, 5877, 6347, 6375, 6480, 6550, 6727, 7369, 7655, 7954, 8050, 10008, 10659, 11050, 11570, 11778, 12395, 13034, 13390, 14364, 14704, 21172, 21229 |
|
Px + 3Py + 8P”z |
47 |
2, 6, 7, 10, 14, 18, 26, 29, 34, 39, 42, 47, 63, 69, 80, 84, 102, 119, 130, 138, 139, 149, 159, 182, 257, 294, 322, 414, 417, 439, 467, 615, 724, 768, 780, 879, 966, 970, 1029, 1239, 1389, 1519, 1559, 2254, 2679, 2720, 21229 |
|
Px + 3P”y + 8Pz |
25 |
2, 10, 17, 31, 32, 39, 42, 63, 69, 82, 89, 126, 143, 149, 207, 208, 257, 260, 427, 467, 507, 519, 600, 845, 966 |
|
P”x + 3Py + 8Pz |
36 |
6, 14, 19, 21, 31, 32, 39, 53, 61, 64, 82, 119, 124, 130, 138, 142, 159, 207, 235, 259, 273, 274, 294, 314, 322, 325, 339, 362, 382, 427, 649, 679, 729, 795, 1354, 6727 |
|
Px + 3P”y + 8P”z |
5 |
2, 10, 39, 69, 149 |
|
P”x + 3Py + 8P”z |
3 |
6, 14, 39 |
|
P”x + 3P”y + 8Pz |
3 |
39, 82, 207 |
|
Px + 3Py + 9Pz |
252 |
2, 6, 7, 11, 18, 19, 23, 26, 28, 30, 32, 33, 39, 40, 42, 43, 52, 55, 56, 62, 64, 68, 69, 74, 77, 84, 89, 90, 91, 98, 100, 102, 121, 122, 125, 131, 134, 138, 139, 142, 147, 161, 164, 168, 170, 171, 180, 182, 187, 189, 194, 207, 208, 216, 217, 227, 229, 230, 237, 238, 243, 263, 272, 274, 275, 278, 282, 294, 300, 301, 309, 310, 314, 317, 341, 344, 348, 349, 357, 359, 367, 371, 380, 383, 387, 394, 399, 404, 414, 415, 417, 426, 427, 433, 439, 446, 450, 458, 472, 482, 493, 516, 518, 524, 534, 545, 567, 581, 583, 601, 609, 616, 622, 632, 649, 653, 662, 664, 677, 692, 693, 702, 712, 716, 723, 744, 754, 768, 769, 771, 779, 799, 802, 816, 819, 839, 874, 881, 909, 924, 930, 947, 958, 965, 977, 1008, 1017, 1024, 1029, 1044, 1063, 1093, 1134, 1147, 1156, 1164, 1169, 1187, 1231, 1238, 1291, 1294, 1301, 1302, 1337, 1339, 1362, 1374, 1379, 1403, 1413, 1442, 1472, 1499, 1563, 1572, 1615, 1674, 1693, 1703, 1764, 1812, 1889, 1909, 1983, 2072, 2074, 2126, 2130, 2168, 2267, 2354, 2539, 2564, 2727, 2729, 2760, 2779, 2862, 2900, 2947, 2967, 2989, 3117, 3122, 3142, 3219, 3238, 3292, 3393, 3429, 3458, 3639, 3835, 3904, 3922, 4307, 4514, 4641, 4867, 4927, 5014, 5262, 5550, 5641, 5763, 5772, 6029, 6067, 6348, 6517, 7047, 7868, 7933, 8559, 8577, 9312, 9478, 9655, 10534, 12074, 12858, 12952, 13559, 15399, 15428, 16102, 16142, 17761, 23494, 27869, 36696 |
|
Px + 3Py + 9P”z |
38 |
2, 6, 7, 11, 28, 32, 39, 42, 52, 62, 74, 77, 102, 122, 131, 142, 161, 182, 187, 207, 317, 349, 359, 383, 472, 482, 493, 534, 609, 754, 768, 779, 947, 1029, 1337, 1472, 2072, 2947 |
|
Px + 3P”y + 9Pz |
27 |
2, 19, 23, 32, 39, 40, 62, 64, 68, 69, 74, 77, 84, 131, 139, 161, 189, 194, 359, 472, 482, 524, 692, 819, 1063, 1231, 1339 |
|
P”x + 3Py + 9Pz |
40 |
6, 23, 28, 32, 33, 42, 56, 84, 91, 98, 139, 142, 147, 161, 194, 229, 243, 294, 309, 314, 359, 394, 414, 433, 439, 493, 518, 567, 601, 744, 768, 779, 839, 977, 1017, 1231, 1294, 2779, 2947, 3639 |
|
Px + 3P”y + 9P”z |
4 |
2, 32, 62, 77 |
|
P”x + 3Py + 9P”z |
3 |
6, 32, 42 |
|
P”x + 3P”y + 9Pz |
3 |
84, 139, 194 |
Tutte queste rappresentazioni sono state verificate per gli interi fino a 1011.