Place Value 732 268 112: Fill & Download for Free

If I had to bet, I’d say yes, but the fact is that there isn’t one for [math]n \leq 5000000[/math]. If you asked me if there are infinitely many such [math]n[/math], I’ll be on the fence. It is definitely very rare for large [math]n[/math], but so are twin primes, which doesn’t mean there aren’t infinitely many of them.That said, there are 82 numbers between 9598 and 5 million for which the sum is exactly 37:9739, 9742, 9824, 9895, 10078, 10301, 10453, 10624, 10627, 10643, 10771, 10882, 10883, 10924, 10927, 10928, 11156, 11482, 11483, 11485, 11488, 11489, 11663, 11896, 11897, 12146, 12148, 12149, 12156, 12157, 12472, 12598, 12884, 12997, 13001, 13003, 13301, 13303, 13304, 13436, 14071, 14072, 15116, 15284, 15296, 15373, 15439, 15667, 15668, 15874, 16216, 16217, 16219, 16916, 17482, 17491, 19438, 20333, 20476, 21211, 21212, 21374, 22024, 22798, 23189, 24683, 26944, 27031, 28867, 29122, 29396, 36899, 45271, 53891, 60611, 63689, 87211, 146512, 169304, 241873, 292469, 1361123 Most of them (77) are less than 100,000; then there’s 4 between 100,000 and 300,000, and just 1 between 300,000 and 5 million. That sole exception (1,361,123) is what gives us hope that the same type of exception may happen, even for 36, down the line, even though the probability of that happening becomes vanishingly small.Actually, looking at the number of prime divisors between 1,361,123 and 1,361,138 we have1, 3, 2, 3, 2, 2, 3, 5, 1, 2, 2, 2, 2, 4, 1, 2 i.e. if it weren’t for this annoying [math]1,361,130 = 2 \cdot 3 \cdot 5 \cdot 59 \cdot 769[/math], we could’ve made the 36 mark!In general, this is the distribution of the sums for n between 9598 and 5 million:36: 1 37: 82 38: 488 39: 2548 40: 8734 41: 25849 42: 65264 43: 146314 44: 282755 45: 472818 46: 669709 47: 802023 48: 809589 49: 689436 50: 490759 51: 292782 52: 144572 53: 59627 54: 20060 55: 5540 56: 1215 57: 201 58: 33 59: 3 The absolute lower minimum for this sum is 8 (factors 2) + 5 (factors 3) + 3 (factors 5) + 2 (7) + 1 (11) + 1 (13) = 20, which seems to give a lot of space to accomodate additional primes and still be at 36 (we can allow one new prime per number, in average), but, for some reason, the cut at 9598 is very abrupt. This is the list of the n’s with sum = 36 from 2 to 5 million:315, 363, 399, 402, 403, 406, 414, 415, 420, 423, 425, 426, 427, 429, 450, 455, 459, 460, 461, 465, 469, 470, 480, 483, 492, 504, 505, 510, 513, 515, 516, 517, 518, 519, 520, 522, 524, 531, 540, 543, 544, 547, 548, 551, 552, 555, 558, 559, 560, 579, 581, 582, 583, 585, 588, 594, 595, 597, 600, 601, 602, 603, 605, 606, 608, 623, 624, 627, 630, 635, 637, 642, 643, 644, 648, 649, 652, 653, 659, 663, 665, 675, 679, 680, 681, 682, 684, 685, 687, 688, 689, 692, 695, 696, 704, 706, 707, 708, 710, 711, 713, 726, 727, 728, 729, 731, 732, 733, 736, 737, 738, 739, 744, 745, 747, 748, 750, 752, 753, 755, 756, 759, 762, 766, 772, 773, 781, 783, 784, 785, 786, 788, 794, 797, 799, 800, 806, 807, 814, 815, 816, 819, 822, 844, 848, 849, 850, 854, 856, 857, 859, 864, 866, 868, 869, 879, 880, 883, 886, 905, 906, 908, 911, 913, 916, 917, 918, 922, 925, 927, 928, 929, 932, 941, 946, 947, 949, 950, 952, 956, 957, 960, 965, 966, 972, 989, 992, 994, 995, 996, 997, 998, 1003, 1007, 1010, 1011, 1012, 1014, 1015, 1021, 1027, 1028, 1036, 1040, 1041, 1042, 1043, 1044, 1045, 1051, 1069, 1072, 1073, 1074, 1075, 1078, 1080, 1085, 1089, 1093, 1094, 1111, 1112, 1115, 1117, 1121, 1122, 1138, 1139, 1141, 1142, 1148, 1149, 1150, 1151, 1171, 1172, 1174, 1178, 1179, 1180, 1181, 1186, 1187, 1191, 1192, 1193, 1198, 1199, 1200, 1211, 1212, 1216, 1219, 1224, 1225, 1226, 1237, 1244, 1247, 1248, 1268, 1271, 1274, 1278, 1283, 1286, 1288, 1294, 1310, 1315, 1319, 1321, 1346, 1358, 1359, 1360, 1362, 1363, 1364, 1371, 1372, 1373, 1423, 1424, 1426, 1427, 1429, 1431, 1437, 1443, 1451, 1453, 1457, 1458, 1466, 1478, 1486, 1487, 1489, 1499, 1521, 1522, 1523, 1531, 1556, 1559, 1561, 1562, 1563, 1566, 1567, 1568, 1606, 1608, 1609, 1611, 1614, 1615, 1617, 1619, 1655, 1656, 1657, 1661, 1663, 1667, 1678, 1681, 1682, 1685, 1686, 1687, 1688, 1706, 1709, 1711, 1712, 1713, 1717, 1719, 1720, 1721, 1723, 1744, 1747, 1750, 1752, 1753, 1772, 1774, 1777, 1786, 1787, 1808, 1849, 1852, 1853, 1856, 1858, 1859, 1860, 1866, 1868, 1871, 1939, 1979, 1982, 1984, 1985, 1993, 1996, 1997, 2014, 2016, 2018, 2024, 2038, 2047, 2074, 2092, 2096, 2098, 2173, 2174, 2186, 2192, 2203, 2206, 2236, 2238, 2239, 2293, 2294, 2296, 2298, 2299, 2300, 2301, 2303, 2304, 2347, 2371, 2381, 2388, 2419, 2420, 2422, 2423, 2426, 2462, 2557, 2587, 2588, 2591, 2633, 2636, 2641, 2643, 2647, 2648, 2719, 2731, 2738, 2788, 2796, 2797, 2851, 2887, 2888, 2894, 3083, 3202, 3203, 3316, 3319, 3361, 3446, 3448, 3449, 3453, 3456, 3571, 3631, 3746, 3917, 3986, 4043, 4049, 4096, 4124, 4217, 4348, 4349, 4478, 4778, 4786, 4787, 4790, 4791, 4798, 4903, 4904, 4996, 4999, 5086, 5087, 5093, 5227, 5314, 5407, 5435, 5569, 6073, 6074, 6119, 6121, 6646, 6647, 6648, 6649, 6856, 6857, 7639, 8419, 8563, 8999, 9598 As we’ve seen, the cut for 37 is much less abrupt. But, as we’ve also seen, no number below 37 is reached again up to n = 5 million. For reference, these are the lists for smaller sums:35:195, 207, 252, 258, 272, 273, 285, 294, 295, 297, 300, 308, 318, 321, 327, 330, 357, 360, 362, 364, 365, 366, 370, 384, 385, 390, 393, 395, 396, 398, 400, 401, 404, 407, 408, 410, 411, 412, 413, 416, 417, 422, 424, 428, 430, 432, 433, 434, 435, 440, 441, 447, 451, 453, 454, 456, 458, 463, 464, 466, 467, 471, 481, 482, 489, 490, 491, 493, 495, 503, 506, 507, 514, 521, 523, 526, 527, 528, 530, 532, 537, 538, 539, 541, 542, 553, 557, 561, 567, 568, 569, 571, 572, 573, 574, 575, 576, 577, 578, 584, 587, 589, 590, 591, 596, 598, 604, 607, 610, 612, 614, 616, 618, 620, 621, 622, 625, 626, 629, 631, 632, 633, 634, 638, 639, 640, 641, 646, 661, 662, 664, 666, 667, 669, 670, 676, 677, 683, 686, 691, 694, 697, 698, 709, 712, 714, 720, 723, 725, 742, 746, 749, 751, 754, 757, 758, 760, 763, 764, 782, 787, 796, 808, 809, 817, 818, 820, 821, 823, 824, 826, 828, 830, 831, 832, 833, 834, 835, 836, 837, 839, 841, 842, 851, 852, 853, 862, 863, 871, 873, 874, 875, 876, 878, 881, 904, 907, 914, 919, 926, 953, 958, 959, 961, 962, 963, 964, 969, 991, 1004, 1006, 1009, 1013, 1017, 1019, 1024, 1037, 1038, 1039, 1047, 1049, 1076, 1079, 1081, 1083, 1084, 1086, 1087, 1088, 1114, 1124, 1201, 1202, 1213, 1214, 1222, 1249, 1276, 1277, 1279, 1282, 1289, 1291, 1292, 1293, 1312, 1313, 1314, 1316, 1317, 1318, 1361, 1368, 1370, 1432, 1433, 1436, 1438, 1439, 1445, 1446, 1447, 1564, 1565, 1607, 1612, 1613, 1616, 1618, 1654, 1684, 1718, 1751, 1861, 1862, 1863, 1865, 1867, 2017, 2048, 2194, 2237, 2297, 2302, 2389, 2642, 2644, 2789, 3452, 3454, 3455, 4789 34:174, 175, 180, 198, 200, 201, 202, 203, 204, 205, 206, 208, 209, 210, 216, 217, 219, 220, 225, 245, 246, 251, 253, 255, 259, 260, 261, 264, 265, 270, 271, 275, 276, 279, 280, 282, 284, 286, 290, 291, 293, 296, 298, 299, 301, 303, 304, 305, 306, 307, 309, 310, 312, 314, 316, 317, 319, 320, 322, 325, 326, 328, 329, 333, 335, 336, 339, 340, 342, 345, 350, 351, 354, 355, 356, 359, 361, 367, 368, 369, 371, 372, 375, 376, 377, 378, 380, 381, 387, 388, 389, 391, 392, 394, 397, 409, 418, 419, 421, 431, 436, 437, 438, 442, 444, 445, 448, 449, 452, 457, 472, 473, 474, 475, 479, 484, 485, 486, 487, 488, 494, 496, 501, 502, 508, 509, 511, 512, 529, 533, 534, 535, 536, 554, 556, 564, 565, 566, 586, 592, 593, 599, 611, 613, 617, 619, 628, 647, 668, 671, 672, 674, 715, 717, 721, 722, 724, 743, 761, 827, 829, 838, 872, 877, 967, 968, 970, 971, 1016, 1018, 1046, 1048, 1082, 1123, 1223, 1366, 1367, 1369, 1444, 1864 33:105, 129, 130, 132, 140, 141, 150, 153, 165, 168, 170, 171, 172, 173, 176, 177, 181, 182, 183, 194, 196, 197, 199, 213, 215, 218, 221, 222, 223, 224, 226, 227, 228, 230, 231, 234, 240, 244, 247, 249, 250, 254, 257, 262, 263, 266, 267, 274, 277, 278, 281, 283, 287, 288, 289, 292, 302, 311, 313, 323, 324, 331, 332, 334, 337, 338, 341, 343, 344, 348, 349, 352, 353, 358, 373, 374, 379, 382, 383, 386, 439, 443, 446, 476, 477, 478, 497, 498, 500, 562, 563, 673, 716, 718, 719 32:84, 90, 102, 104, 108, 109, 110, 111, 126, 128, 131, 133, 135, 138, 139, 142, 143, 144, 145, 146, 147, 151, 152, 154, 155, 156, 159, 160, 161, 162, 163, 164, 167, 169, 178, 179, 184, 185, 186, 189, 190, 192, 193, 211, 212, 214, 229, 232, 233, 235, 237, 238, 243, 248, 256, 268, 269, 346, 347, 499 31:55, 63, 65, 75, 76, 77, 78, 80, 81, 82, 83, 85, 87, 91, 99, 100, 101, 103, 106, 107, 112, 114, 115, 117, 118, 119, 120, 123, 125, 127, 134, 136, 137, 148, 149, 157, 158, 166, 187, 188, 191, 236, 239, 241, 242 30:51, 54, 56, 57, 60, 62, 64, 66, 69, 70, 72, 73, 74, 79, 86, 88, 89, 92, 93, 95, 96, 97, 98, 113, 116, 121, 122, 124 29:30, 33, 42, 45, 48, 50, 52, 53, 58, 59, 61, 67, 68, 71, 94 28:27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 46, 47, 49 27:20, 21, 24, 25, 26 26:15, 18, 19, 22, 23 25:9, 10, 11, 12, 13, 14, 16, 17 24:6, 7, 8 23:5 22:3, 4 21:2 So, the question is, what are the values of [math]S[/math] for which there are only finitely many [math]n[/math] such that the sum is exactly [math]S[/math]? We know that all S<20 have that property. Can we prove that no n>2 has sum = 21?As we’ve seen, that means than all numbers between n and n+15 are only divisible by 2,3,5, 7, 11, 13 and one other prime. More than that, the 3’s, 5’s and 7’s cannot appear more than the bare minimum of times. So, let’s try and cover all the numbers from n to n+15 with our primes.To start, with the factor 2 we can cover 8 numbers, regardless of whether we make n even or odd. Let’s make it even, then:Now, we need to place the 3’s. If we make n multiple of 3, we’d have 6 3’s instead of 5, so that’s forbidden. If we make n+2 multiple of 3, we’d only cover two additional numbers - n+5 and n+11 -, whereas making n+1 multiple of 3 we cover three more numbers:So we covered 11 numbers so far, of our 16, and that’s the best we can do at this point. To place the 5’s, again, we don’t want to make n multiple of 5, or we’d have four 5’s instead of 3. In any other position, we’d only cover at most one additional number, and two numbers already covered by 2 or 3 [íf we put 5 at n+1, n+3 or n+4], or no additional number at all [if we put it at n+2].This is one of the options. So far, we’ve covered 12 numbers, and this is optimal.For the 7’s, we don’t want to put them at n+0 nor n+1, and now we will have 2 numbers, one of which will be necessarily even. Hence, we can only cover one additional number.Now, we have at least 3 numbers (marked in yellow) that haven’t been covered, and we only have 11 and 13 left. So, let’s cover two of them with the remaining primesActually, we know that we’ll always have either n or n+15 left open. So we need a new prime [math]p[/math] to divide the remaining number.But now n cannot be divisible by no other prime than 2 (since p>15, and p divides n+15, p cannot divide n, and we’ve already seen that none of the other, smaller primes do).And now it’s easy to get to contradictions, regardless of the configuration we selected, unless n = 2. For instance, in this configuration, we have n and n+2 both powers of 2, which imply n = 2. But, more generally, we’ve been almost forced to have 3 | (n+1), which means [math]n = 2^k[/math] with k odd. If k > 1, it has an odd prime power q, and n+1 would be multiple of [math]2^q + 1[/math], and so on. It’s just a finite number of combinations to check and we’re done.A couple of big outliers, that are close, but no cigar:S(4,033,131,103) = 39S(92,509,176,761) = 41S(121,050,191,263) = 42So, we know that, for small values of [math]S[/math], the problem is constrained enough to allow only for finitely many solutions. What’s the turning point, if any? Is it true for all S? Is it at S=37 that we start having infinitely many solutions? Wouldn’t it be wonderful if it were at S=42!?I tried to find some reference to that problem, but I couldn’t. Damn, I cannot even think of an argument why S = one googol would necessarily have infinitely many solutions. So, in summary, I’ve written a pretty long answer just to say “I don’t know”.