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Repunit

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers.[note 1]

A repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes.

Thus, the number Rn(b) consists of n copies of the digit 1 in base-b representation. The first two repunits base-b for n = 1 and n = 2 are

In particular, the decimal (base-10) repunits that are often referred to as simply repunits are defined as

Thus, the number Rn = Rn(10) consists of n copies of the digit 1 in base 10 representation. The sequence of repunits base-10 starts with

Thus, the number Rn(2) consists of n copies of the digit 1 in base-2 representation. In fact, the base-2 repunits are the well-known Mersenne numbers Mn = 2n − 1, they start with

(Prime factors colored red means "new factors", i. e. the prime factor divides Rn but does not divide Rk for all k < n) (sequence A102380 in the OEIS)[2]

The definition of repunits was motivated by recreational mathematicians looking for prime factors of such numbers.

It is easy to show that if n is divisible by a, then Rn(b) is divisible by Ra(b):

where is the cyclotomic polynomial and d ranges over the divisors of n. For p prime,

which has the expected form of a repunit when x is substituted with b.

For example, 9 is divisible by 3, and thus R9 is divisible by R3—in fact, 111111111 = 111 · 1001001. The corresponding cyclotomic polynomials and are and , respectively. Thus, for Rn to be prime, n must necessarily be prime, but it is not sufficient for n to be prime. For example, R3 = 111 = 3 · 37 is not prime. Except for this case of R3, p can only divide Rn for prime n if p = 2kn + 1 for some k.

Rn is prime for n = 2, 19, 23, 317, 1031, ... (sequence A004023 in OEIS). R49081 and R86453 are probably prime. On April 3, 2007 Harvey Dubner (who also found R49081) announced that R109297 is a probable prime.[3] He later announced there are no others from R86453 to R200000.[4] On July 15, 2007 Maksym Voznyy announced R270343 to be probably prime,[5] along with his intent to search to 400000. As of November 2012, all further candidates up to R2500000 have been tested, but no new probable primes have been found so far.

It has been conjectured that there are infinitely many repunit primes[6] and they seem to occur roughly as often as the prime number theorem would predict: the exponent of the Nth repunit prime is generally around a fixed multiple of the exponent of the (N−1)th.

The prime repunits are a trivial subset of the permutable primes, i.e., primes that remain prime after any permutation of their digits.

The only base-4 repunit prime is 5 (). , and 3 always divides when n is odd and when n is even. For n greater than 2, both and are greater than 3, so removing the factor of 3 still leaves two factors greater than 1. Therefore, the number cannot be prime.

The only base-8 repunit prime is 73 (). , and 7 divides when n is not divisible by 3 and when n is a multiple of 3.

There are no base-9 repunit primes. , and both and are even and greater than 4.

bases such that is prime (only lists positive bases) OEIS sequence 2 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270, 276, 280, 282, 292, 306, 310, 312, 316, 330, 336, 346, 348, 352, 358, 366, 372, 378, 382, 388, 396, 400, 408, 418, 420, 430, 432, 438, 442, 448, 456, 460, 462, 466, 478, 486, 490, 498, 502, 508, 520, 522, 540, 546, 556, 562, 568, 570, 576, 586, 592, 598, 600, 606, 612, 616, 618, 630, 640, 642, 646, 652, 658, 660, 672, 676, 682, 690, 700, 708, 718, 726, 732, 738, 742, 750, 756, 760, 768, 772, 786, 796, 808, 810, 820, 822, 826, 828, 838, 852, 856, 858, 862, 876, 880, 882, 886, 906, 910, 918, 928, 936, 940, 946, 952, 966, 970, 976, 982, 990, 996, ... A006093 3 2, 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119, 131, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 167, 168, 173, 176, 188, 189, 192, 194, 203, 206, 209, 215, 218, 231, 236, 245, 246, 266, 272, 278, 279, 287, 288, 290, 293, 309, 314, 329, 332, 336, 342, 344, 348, 351, 357, 369, 378, 381, 383, 392, 395, 398, 402, 404, 405, 414, 416, 426, 434, 435, 447, 453, 455, 456, 476, 489, 495, 500, 512, 518, 525, 530, 531, 533, 537, 540, 551, 554, 560, 566, 567, 572, 579, 582, 584, 603, 605, 609, 612, 621, 624, 626, 635, 642, 644, 668, 671, 677, 686, 696, 701, 720, 726, 728, 735, 743, 747, 755, 761, 762, 768, 773, 782, 785, 792, 798, 801, 812, 818, 819, 825, 827, 836, 839, 846, 855, 857, 860, 864, 875, 878, 890, 894, 897, 899, 911, 915, 918, 920, 927, 950, 959, 960, 969, 974, 981, 987, 990, 992, 993, ... A002384 5 2, 7, 12, 13, 17, 22, 23, 24, 28, 29, 30, 40, 43, 44, 50, 62, 63, 68, 73, 74, 77, 79, 83, 85, 94, 99, 110, 117, 118, 120, 122, 127, 129, 134, 143, 145, 154, 162, 164, 165, 172, 175, 177, 193, 198, 204, 208, 222, 227, 239, 249, 254, 255, 260, 263, 265, 274, 275, 277, 285, 288, 292, 304, 308, 327, 337, 340, 352, 359, 369, 373, 393, 397, 408, 414, 417, 418, 437, 439, 448, 457, 459, 474, 479, 490, 492, 495, 503, 505, 514, 519, 528, 530, 538, 539, 540, 550, 557, 563, 567, 568, 572, 579, 594, 604, 617, 637, 645, 650, 662, 679, 694, 699, 714, 728, 745, 750, 765, 770, 772, 793, 804, 805, 824, 837, 854, 860, 864, 868, 880, 890, 919, 942, 954, 967, 968, 974, 979, ... A049409 7 2, 3, 5, 6, 13, 14, 17, 26, 31, 38, 40, 46, 56, 60, 61, 66, 68, 72, 73, 80, 87, 89, 93, 95, 115, 122, 126, 128, 146, 149, 156, 158, 160, 163, 180, 186, 192, 203, 206, 208, 220, 221, 235, 237, 238, 251, 264, 266, 280, 282, 290, 294, 300, 303, 320, 341, 349, 350, 353, 363, 381, 395, 399, 404, 405, 417, 418, 436, 438, 447, 450, 461, 464, 466, 478, 523, 531, 539, 548, 560, 583, 584, 591, 599, 609, 611, 622, 646, 647, 655, 657, 660, 681, 698, 700, 710, 717, 734, 760, 765, 776, 798, 800, 802, 805, 822, 842, 856, 863, 870, 878, 899, 912, 913, 926, 927, 931, 940, 941, 942, 947, 959, 984, 998, ... A100330 11 5, 17, 20, 21, 30, 53, 60, 86, 137, 172, 195, 212, 224, 229, 258, 268, 272, 319, 339, 355, 365, 366, 389, 390, 398, 414, 467, 480, 504, 534, 539, 543, 567, 592, 619, 626, 654, 709, 735, 756, 766, 770, 778, 787, 806, 812, 874, 943, 973, ... A162862 13 2, 3, 5, 7, 34, 37, 43, 59, 72, 94, 98, 110, 133, 149, 151, 159, 190, 207, 219, 221, 251, 260, 264, 267, 282, 286, 291, 319, 355, 363, 373, 382, 397, 398, 402, 406, 408, 412, 436, 442, 486, 489, 507, 542, 544, 552, 553, 582, 585, 592, 603, 610, 614, 634, 643, 645, 689, 708, 720, 730, 744, 769, 772, 806, 851, 853, 862, 882, 912, 928, 930, 952, 968, 993, ... A217070 17 2, 11, 20, 21, 28, 31, 55, 57, 62, 84, 87, 97, 107, 109, 129, 147, 149, 157, 160, 170, 181, 189, 191, 207, 241, 247, 251, 274, 295, 297, 315, 327, 335, 349, 351, 355, 364, 365, 368, 379, 383, 410, 419, 423, 431, 436, 438, 466, 472, 506, 513, 527, 557, 571, 597, 599, 614, 637, 653, 656, 688, 708, 709, 720, 740, 762, 835, 836, 874, 974, 976, 980, 982, 986, ... A217071 19 2, 10, 11, 12, 14, 19, 24, 40, 45, 46, 48, 65, 66, 67, 75, 85, 90, 103, 105, 117, 119, 137, 147, 164, 167, 179, 181, 205, 220, 235, 242, 253, 254, 263, 268, 277, 303, 315, 332, 337, 366, 369, 370, 389, 399, 404, 424, 431, 446, 449, 480, 481, 506, 509, 521, 523, 531, 547, 567, 573, 581, 622, 646, 651, 673, 736, 768, 787, 797, 807, 810, 811, 817, 840, 846, 857, 867, 869, 870, 888, 899, 902, 971, 988, 990, 992, ... A217072 23 10, 40, 82, 113, 127, 141, 170, 257, 275, 287, 295, 315, 344, 373, 442, 468, 609, 634, 646, 663, 671, 710, 819, 834, 857, 884, 894, 904, 992, 997, ... A217073 29 6, 40, 65, 70, 114, 151, 221, 229, 268, 283, 398, 451, 460, 519, 554, 587, 627, 628, 659, 687, 699, 859, 884, 915, 943, 974, 986, ... A217074 31 2, 14, 19, 31, 44, 53, 71, 82, 117, 127, 131, 145, 177, 197, 203, 241, 258, 261, 276, 283, 293, 320, 325, 379, 387, 388, 406, 413, 461, 462, 470, 486, 491, 534, 549, 569, 582, 612, 618, 639, 696, 706, 723, 746, 765, 767, 774, 796, 802, 877, 878, 903, 923, 981, 991, 998, ... A217075 37 61, 77, 94, 97, 99, 113, 126, 130, 134, 147, 161, 172, 187, 202, 208, 246, 261, 273, 285, 302, 320, 432, 444, 503, 523, 525, 563, 666, 680, 709, 740, 757, 787, 902, 962, 964, 969, ... A217076 41 14, 53, 55, 58, 71, 76, 82, 211, 248, 271, 296, 316, 430, 433, 439, 472, 545, 553, 555, 596, 663, 677, 682, 746, 814, 832, 885, 926, 947, 959, ... A217077 43 15, 21, 26, 86, 89, 114, 123, 163, 180, 310, 332, 377, 409, 438, 448, 457, 477, 526, 534, 556, 586, 612, 653, 665, 690, 692, 709, 760, 783, 803, 821, 848, 877, 899, 909, 942, 981, ... A217078 47 5, 17, 19, 55, 62, 75, 89, 98, 99, 132, 172, 186, 197, 220, 268, 278, 279, 288, 439, 443, 496, 579, 583, 587, 742, 777, 825, 911, 966, ... A217079 53 24, 45, 60, 165, 235, 272, 285, 298, 307, 381, 416, 429, 623, 799, 858, 924, 929, 936, ... A217080 59 19, 70, 102, 116, 126, 188, 209, 257, 294, 359, 451, 461, 468, 470, 638, 653, 710, 762, 766, 781, 824, 901, 939, 964, 995, ... A217081 61 2, 19, 69, 88, 138, 155, 205, 234, 336, 420, 425, 455, 470, 525, 555, 561, 608, 626, 667, 674, 766, 779, 846, 851, 937, 971, 998, ... A217082 67 46, 122, 238, 304, 314, 315, 328, 332, 346, 372, 382, 426, 440, 491, 496, 510, 524, 528, 566, 638, 733, 826, ... A217083 71 3, 6, 17, 24, 37, 89, 132, 374, 387, 402, 421, 435, 453, 464, 490, 516, 708, 736, 919, 947, 981, ... A217084 73 11, 15, 75, 114, 195, 215, 295, 335, 378, 559, 566, 650, 660, 832, 871, 904, 966, ... A217085 79 22, 112, 140, 158, 170, 254, 271, 330, 334, 354, 390, 483, 528, 560, 565, 714, 850, 888, 924, 929, 933, 935, 970, ... A217086 83 41, 146, 386, 593, 667, 688, 906, 927, 930, ... A217087 89 2, 114, 159, 190, 234, 251, 436, 616, 834, 878, ... A217088 97 12, 90, 104, 234, 271, 339, 420, 421, 428, 429, 464, 805, 909, 934, ... A217089 101 22, 78, 164, 302, 332, 359, 387, 428, 456, 564, 617, 697, 703, 704, 785, 831, 979, ... 103 3, 52, 345, 392, 421, 472, 584, 617, 633, 761, 767, 775, 785, 839, ... 107 2, 19, 61, 68, 112, 157, 219, 349, 677, 692, 700, 809, 823, 867, 999, ... 109 12, 57, 72, 79, 89, 129, 158, 165, 239, 240, 260, 277, 313, 342, 421, 445, 577, 945, ... 113 86, 233, 266, 299, 334, 492, 592, 641, 656, 719, 946, ... 127 2, 5, 6, 47, 50, 126, 151, 226, 250, 401, 427, 473, 477, 486, 497, 585, 624, 644, 678, 685, 687, 758, 896, 897, 936, ... 131 7, 493, 567, 591, 593, 613, 764, 883, 899, 919, 953, ... 137 13, 166, 213, 355, 586, 669, 707, 768, 833, ... 139 11, 50, 221, 415, 521, 577, 580, 668, 717, 720, 738, 902, ... 149 5, 7, 68, 79, 106, 260, 319, 502, 550, 779, 855, ... 151 29, 55, 57, 160, 176, 222, 255, 364, 427, 439, 642, 660, 697, 863, ... 157 56, 71, 76, 181, 190, 317, 338, 413, 426, 609, 694, 794, 797, 960, ... 163 30, 62, 118, 139, 147, 291, 456, 755, 834, 888, 902, 924, ... 167 44, 45, 127, 175, 182, 403, 449, 453, 476, 571, 582, 700, 749, 764, 929, 957, ... 173 60, 62, 139, 141, 303, 313, 368, 425, 542, 663, ... 179 304, 478, 586, 942, 952, 975, ... 181 5, 37, 171, 427, 509, 571, 618, 665, 671, 786, ... 191 74, 214, 416, 477, 595, 664, 699, 712, 743, 924, ... 193 118, 301, 486, 554, 637, 673, 736, ... 197 33, 236, 248, 262, 335, 363, 388, 593, 763, 813, ... 199 156, 362, 383, 401, 442, 630, 645, 689, 740, 921, 936, 944, 983, 988, ... 211 46, 57, 354, 478, 539, 581, 653, 829, 835, 977, ... 223 183, 186, 219, 221, 661, 749, 905, 914, ... 227 72, 136, 235, 240, 251, 322, 350, 500, 523, 556, 577, 671, 688, 743, 967, ... 229 606, 725, 754, 858, 950, ... 233 602, ... 239 223, 260, 367, 474, 564, 862, ... 241 115, 163, 223, 265, 270, 330, 689, 849, ... 251 37, 246, 267, 618, 933, ... 257 52, 78, 435, 459, 658, 709, ... 263 104, 131, 161, 476, 494, 563, 735, 842, 909, 987, ... 269 41, 48, 294, 493, 520, 812, 843, ... 271 6, 21, 186, 201, 222, 240, 586, 622, 624, ... 277 338, 473, 637, 940, 941, 978, ... 281 217, 446, 606, 618, 790, 864, ... 283 13, 197, 254, 288, 323, 374, 404, 943, ... 293 136, 388, 471, ...

Smallest prime such that is prime are (start with , 0 if no such exists, question mark if this term is currently unknown)

numbers such that is prime (some large terms are only corresponding to probable primes, these are checked up to 100000) OEIS sequence −50 1153, 26903, 56597, ... A309413 −49 7, 19, 37, 83, 1481, 12527, 20149, ... A237052 −48 2*, 5, 17, 131, 84589, ... A236530 −47 5, 19, 23, 79, 1783, 7681, ... A236167 −46 7, 23, 59, 71, 107, 223, 331, 2207, 6841, 94841, ... A235683 −45 103, 157, 37159, ... A309412 −44 2*, 7, 41233, ... A309411 −43 5, 7, 19, 251, 277, 383, 503, 3019, 4517, 9967, 29573, ... A231865 −42 2*, 3, 709, 1637, 17911, 127609, 172663, ... A231604 −41 17, 691, 113749, ... A309410 −40 53, 67, 1217, 5867, 6143, 11681, 29959, ... A229663 −39 3, 13, 149, 15377, ... A230036 −38 2*, 5, 167, 1063, 1597, 2749, 3373, 13691, 83891, 131591, ... A229524 −37 5, 7, 2707, 163193, ... A309409 −36 31, 191, 257, 367, 3061, 110503, ... A229145 −35 11, 13, 79, 127, 503, 617, 709, 857, 1499, 3823, 135623, ... A185240 −34 3, 294277, ... −33 5, 67, 157, 12211, ... A185230 −32 2* (no others) −31 109, 461, 1061, 50777, ... A126856 −30 2*, 139, 173, 547, 829, 2087, 2719, 3109, 10159, 56543, 80599, ... A071382 −29 7, 112153, 151153, ... A291906 −28 3, 19, 373, 419, 491, 1031, 83497, ... A071381 −27 (none) −26 11, 109, 227, 277, 347, 857, 2297, 9043, ... A071380 −25 3, 7, 23, 29, 59, 1249, 1709, 1823, 1931, 3433, 8863, 43201, 78707, ... A057191 −24 2*, 7, 11, 19, 2207, 2477, 4951, ... A057190 −23 11, 13, 67, 109, 331, 587, 24071, 29881, 44053, ... A057189 −22 3, 5, 13, 43, 79, 101, 107, 227, 353, 7393, 50287, ... A057188 −21 3, 5, 7, 13, 37, 347, 17597, 59183, 80761, 210599, 394579, ... A057187 −20 2*, 5, 79, 89, 709, 797, 1163, 6971, 140053, 177967, 393257, ... A057186 −19 17, 37, 157, 163, 631, 7351, 26183, 30713, 41201, 77951, 476929, ... A057185 −18 2*, 3, 7, 23, 73, 733, 941, 1097, 1933, 4651, 481147, ... A057184 −17 7, 17, 23, 47, 967, 6653, 8297, 41221, 113621, 233689, 348259, ... A057183 −16 3, 5, 7, 23, 37, 89, 149, 173, 251, 307, 317, 30197, 1025393, ... A057182 −15 3, 7, 29, 1091, 2423, 54449, 67489, 551927, ... A057181 −14 2*, 7, 53, 503, 1229, 22637, 1091401, ... A057180 −13 3, 11, 17, 19, 919, 1151, 2791, 9323, 56333, 1199467, ... A057179 −12 2*, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, 495953, ... A057178 −11 5, 7, 179, 229, 439, 557, 6113, 223999, 327001, ... A057177 −10 5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, ... A001562 −9 3, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247, 703393, ... A057175 −8 2* (no others) −7 3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653, 1178033, ... A057173 −6 2*, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, 1313371, ... A057172 −5 5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, 1856147, ... A057171 −4 2*, 3 (no others) −3 2*, 3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, 104243, 134227, 152287, 700897, 1205459, ... A007658 −2 3, 4*, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, ..., 13347311, 13372531, ... A000978 2 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, ..., 57885161, ..., 74207281, ..., 77232917, ... A000043 3 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, ... A028491 4 2 (no others) 5 3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, ... A004061 6 2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, ... A004062 7 5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ... A004063 8 3 (no others) 9 (none) 10 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... A004023 11 17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, ... A005808 12 2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543, ... A004064 13 5, 7, 137, 283, 883, 991, 1021, 1193, 3671, 18743, 31751, 101089, ... A016054 14 3, 7, 19, 31, 41, 2687, 19697, 59693, 67421, 441697, ... A006032 15 3, 43, 73, 487, 2579, 8741, 37441, 89009, 505117, 639833, ... A006033 16 2 (no others) 17 3, 5, 7, 11, 47, 71, 419, 4799, 35149, 54919, 74509, ... A006034 18 2, 25667, 28807, 142031, 157051, 180181, 414269, ... A133857 19 19, 31, 47, 59, 61, 107, 337, 1061, 9511, 22051, 209359, ... A006035 20 3, 11, 17, 1487, 31013, 48859, 61403, 472709, ... A127995 21 3, 11, 17, 43, 271, 156217, 328129, ... A127996 22 2, 5, 79, 101, 359, 857, 4463, 9029, 27823, ... A127997 23 5, 3181, 61441, 91943, 121949, ... A204940 24 3, 5, 19, 53, 71, 653, 661, 10343, 49307, 115597, 152783, ... A127998 25 (none) 26 7, 43, 347, 12421, 12473, 26717, ... A127999 27 3 (no others) 28 2, 5, 17, 457, 1423, 115877, ... A128000 29 5, 151, 3719, 49211, 77237, ... A181979 30 2, 5, 11, 163, 569, 1789, 8447, 72871, 78857, 82883, ... A098438 31 7, 17, 31, 5581, 9973, 101111, ... A128002 32 (none) 33 3, 197, 3581, 6871, 183661, ... A209120 34 13, 1493, 5851, 6379, 125101, ... A185073 35 313, 1297, ... 36 2 (no others) 37 13, 71, 181, 251, 463, 521, 7321, 36473, 48157, 87421, 168527, ... A128003 38 3, 7, 401, 449, 109037, ... A128004 39 349, 631, 4493, 16633, 36341, ... A181987 40 2, 5, 7, 19, 23, 29, 541, 751, 1277, ... A128005 41 3, 83, 269, 409, 1759, 11731, ... A239637 42 2, 1319, ... 43 5, 13, 6277, 26777, 27299, 40031, 44773, ... A240765 44 5, 31, 167, 100511, ... A294722 45 19, 53, 167, 3319, 11257, 34351, ... A242797 46 2, 7, 19, 67, 211, 433, 2437, 2719, 19531, ... A243279 47 127, 18013, 39623, ... A267375 48 19, 269, 349, 383, 1303, 15031, ... A245237 49 (none) 50 3, 5, 127, 139, 347, 661, 2203, 6521, ... A245442

* Repunits with negative base and even n are negative. If their absolute value is prime then they are included above and marked with an asterisk. They are not included in the corresponding OEIS sequences.

If b is a perfect power (can be written as mn, with m, n integers, n > 1) differs from 1, then there is at most one repunit in base-b. If n is a prime power (can be written as pr, with p prime, r integer, p, r >0), then all repunit in base-b are not prime aside from Rp and R2. Rp can be either prime or composite, the former examples, b = −216, −128, 4, 8, 16, 27, 36, 100, 128, 256, etc., the latter examples, b = −243, −125, −64, −32, −27, −8, 9, 25, 32, 49, 81, 121, 125, 144, 169, 196, 216, 225, 243, 289, etc., and R2 can be prime (when p differs from 2) only if b is negative, a power of −2, for example, b = −8, −32, −128, −8192, etc., in fact, the R2 can also be composite, for example, b = −512, −2048, −32768, etc. If n is not a prime power, then no base-b repunit prime exists, for example, b = 64, 729 (with n = 6), b = 1024 (with n = 10), and b = −1 or 0 (with n any natural number). Another special situation is b = −4k4, with k positive integer, which has the aurifeuillean factorization, for example, b = −4 (with k = 1, then R2 and R3 are primes), and b = −64, −324, −1024, −2500, −5184, ... (with k = 2, 3, 4, 5, 6, ...), then no base-b repunit prime exists. It is also conjectured that when b is neither a perfect power nor −4k4 with k positive integer, then there are infinity many base-b repunit primes.

A conjecture related to the generalized repunit primes:[11][12] (the conjecture predicts where is the next generalized Mersenne prime, if the conjecture is true, then there are infinitely many repunit primes for all bases )

base-b repunit primes less than N.

Although they were not then known by that name, repunits in base-10 were studied by many mathematicians during the nineteenth century in an effort to work out and predict the cyclic patterns of repeating decimals.[13]

It was found very early on that for any prime p greater than 5, the period of the decimal expansion of 1/p is equal to the length of the smallest repunit number that is divisible by p. Tables of the period of reciprocal of primes up to 60,000 had been published by 1860 and permitted the factorization by such mathematicians as Reuschle of all repunits up to R16 and many larger ones. By 1880, even R17 to R36 had been factored[13] and it is curious that, though Édouard Lucas showed no prime below three million had period nineteen, there was no attempt to test any repunit for primality until early in the twentieth century. The American mathematician Oscar Hoppe proved R19 to be prime in 1916[14] and Lehmer and Kraitchik independently found R23 to be prime in 1929.

Further advances in the study of repunits did not occur until the 1960s, when computers allowed many new factors of repunits to be found and the gaps in earlier tables of prime periods corrected. R317 was found to be a probable prime circa 1966 and was proved prime eleven years later, when R1031 was shown to be the only further possible prime repunit with fewer than ten thousand digits. It was proven prime in 1986, but searches for further prime repunits in the following decade consistently failed. However, there was a major side-development in the field of generalized repunits, which produced a large number of new primes and probable primes.

Since 1999, four further probably prime repunits have been found, but it is unlikely that any of them will be proven prime in the foreseeable future because of their huge size.

The Cunningham project endeavours to document the integer factorizations of (among other numbers) the repunits to base 2, 3, 5, 6, 7, 10, 11, and 12.

D. R. Kaprekar has defined Demlo numbers as concatenation of a left, middle and right part, where the left and right part must be of the same length (up to a possible leading zero to the left) and must add up to a repdigit number, and the middle part may contain any additional number of this repeated digit.[15] They are named after Demlo railway station 30 miles from Bombay on the then G.I.P. Railway, where Kaprekar started investigating them. He calls Wonderful Demlo numbers those of the form 1, 121, 12321, 1234321, ..., 12345678987654321. The fact that these are the squares of the repunits has led some authors to call Demlo numbers the infinite sequence of these,[16] 1, 121, 12321, ..., 12345678987654321, 1234567900987654321, 123456790120987654321, ..., (sequence A002477 in the OEIS), although one can check these are not Demlo numbers for p = 10, 19, 28, ...