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.
A repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes.
Definition
The base-b repunits are defined asThus, the number Rn 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 repunits that are often referred to as simply repunits are defined as
Thus, the number Rn = Rn consists of n copies of the digit 1 in base 10 representation. The sequence of repunits base-10 starts with
Similarly, the repunits base-2 are defined as
Thus, the number Rn 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
Properties
- Any repunit in any base having a composite number of digits is necessarily composite. Only repunits having a prime number of digits might be prime. This is a necessary but not sufficient condition. For example,
- : R35 = = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001,
- If p is an odd prime, then every prime q that divides Rp must be either 1 plus a multiple of 2p, or a factor of b − 1. For example, a prime factor of R29 is 62003 = 1 + 2·29·1069. The reason is that the prime p is the smallest exponent greater than 1 such that q divides bp − 1, because p is prime. Therefore, unless q divides b − 1, p divides the Carmichael function of q, which is even and equal to q − 1.
- Any positive multiple of the repunit Rn contains at least n nonzero digits in base-b.
- Any number x is a two-digit repunit in base x − 1.
- The only known numbers that are repunits with at least 3 digits in more than one base simultaneously are 31 and 8191. The Goormaghtigh conjecture says there are only these two cases.
- Using the pigeon-hole principle it can be easily shown that for relatively prime natural numbers n and b, there exists a repunit in base-b that is a multiple of n. To see this consider repunits R1,...,Rn. Because there are n repunits but only n−1 non-zero residues modulo n there exist two repunits Ri and Rj with 1 ≤ i < j ≤ n such that Ri and Rj have the same residue modulo n. It follows that Rj − Ri has residue 0 modulo n, i.e. is divisible by n. Since Rj − Ri consists of j − i ones followed by i zeroes,. Now n divides the left-hand side of this equation, so it also divides the right-hand side, but since n and b are relatively prime, n must divide Rj−i.
- The Feit–Thompson conjecture is that Rq never divides Rp for two distinct primes p and q.
- Using the Euclidean Algorithm for repunits definition: R1 = 1; Rn = Rn−1 × b + 1, any consecutive repunits Rn−1 and Rn are relatively prime in any base-b for any n.
- If m and n have a common divisor d, Rm and Rn have the common divisor Rd in any base-b for any m and n. That is, the repunits of a fixed base form a strong divisibility sequence. As a consequence, If m and n are relatively prime, Rm and Rn are relatively prime. The Euclidean Algorithm is based on gcd = gcd for m > n. Similarly, using Rm − Rn × bm−n = Rm−n, it can be easily shown that gcd, Rn = gcd, Rn for m > n. Therefore if gcd = d, then gcd, Rn = Rd.
Factorization of decimal repunits
Repunit primes
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 is divisible by Ra:
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.
Decimal repunit primes
Rn is prime for n = 2, 19, 23, 317, 1031,.... R49081 and R86453 are probably prime. On April 3, 2007 Harvey Dubner announced that R109297 is a probable prime. He later announced there are no others from R86453 to R200000. On July 15, 2007 Maksym Voznyy announced R270343 to be probably prime, 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 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 th.
The prime repunits are a trivial subset of the permutable primes, i.e., primes that remain prime after any permutation of their digits.
Particular properties are
- The remainder of Rn modulo 3 is equal to the remainder of n modulo 3. Using 10a ≡ 1 for any a ≥ 0,
n ≡ 1 ⇔ Rn ≡ 1 ⇔ Rn ≡ R1 ≡ 1,
n ≡ 2 ⇔ Rn ≡ 2 ⇔ Rn ≡ R2 ≡ 11.
Therefore, 3 | n ⇔ 3 | Rn ⇔ R3 | Rn.
- The remainder of Rn modulo 9 is equal to the remainder of n modulo 9. Using 10a ≡ 1 for any a ≥ 0,
for 0 ≤ r < 9.
Therefore, 9 | n ⇔ 9 | Rn ⇔ R9 | Rn.
Base 2 repunit primes
Base-2 repunit primes are called Mersenne primes.Base 3 repunit primes
The first few base-3 repunit primes arecorresponding to of
Base 4 repunit primes
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.Base 5 repunit primes
The first few base-5 repunit primes arecorresponding to of
Base 6 repunit primes
The first few base-6 repunit primes arecorresponding to of
Base 7 repunit primes
The first few base-7 repunit primes arecorresponding to of
Base 8 repunit primes
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.Base 9 repunit primes
There are no base-9 repunit primes., and both and are even and greater than 4.Base 11 repunit primes
The first few base-11 repunit primes arecorresponding to of
Base 12 repunit primes
The first few base-12 repunit primes arecorresponding to of
Base 20 repunit primes
The first few base-20 repunit primes arecorresponding to of
Bases such that is prime for prime
Smallest base such that is prime areSmallest base such that is prime are
| bases such that is prime | 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 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,... | |
| 53 | 24, 45, 60, 165, 235, 272, 285, 298, 307, 381, 416, 429, 623, 799, 858, 924, 929, 936,... | |
| 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,... | |
| 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,... | |
| 67 | 46, 122, 238, 304, 314, 315, 328, 332, 346, 372, 382, 426, 440, 491, 496, 510, 524, 528, 566, 638, 733, 826,... | |
| 71 | 3, 6, 17, 24, 37, 89, 132, 374, 387, 402, 421, 435, 453, 464, 490, 516, 708, 736, 919, 947, 981,... | |
| 73 | 11, 15, 75, 114, 195, 215, 295, 335, 378, 559, 566, 650, 660, 832, 871, 904, 966,... | |
| 79 | 22, 112, 140, 158, 170, 254, 271, 330, 334, 354, 390, 483, 528, 560, 565, 714, 850, 888, 924, 929, 933, 935, 970,... | |
| 83 | 41, 146, 386, 593, 667, 688, 906, 927, 930,... | |
| 89 | 2, 114, 159, 190, 234, 251, 436, 616, 834, 878,... | |
| 97 | 12, 90, 104, 234, 271, 339, 420, 421, 428, 429, 464, 805, 909, 934,... | |
| 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,... |
List of repunit primes base
Smallest prime such that is prime areSmallest prime such that is prime are
| numbers such that is prime | OEIS sequence | |
| −50 | 1153, 26903, 56597,... | |
| −49 | 7, 19, 37, 83, 1481, 12527, 20149,... | |
| −48 | 2*, 5, 17, 131, 84589,... | |
| −47 | 5, 19, 23, 79, 1783, 7681,... | |
| −46 | 7, 23, 59, 71, 107, 223, 331, 2207, 6841, 94841,... | |
| −45 | 103, 157, 37159,... | |
| −44 | 2*, 7, 41233,... | |
| −43 | 5, 7, 19, 251, 277, 383, 503, 3019, 4517, 9967, 29573,... | |
| −42 | 2*, 3, 709, 1637, 17911, 127609, 172663,... | |
| −41 | 17, 691, 113749,... | |
| −40 | 53, 67, 1217, 5867, 6143, 11681, 29959,... | |
| −39 | 3, 13, 149, 15377,... | |
| −38 | 2*, 5, 167, 1063, 1597, 2749, 3373, 13691, 83891, 131591,... | |
| −37 | 5, 7, 2707, 163193,... | |
| −36 | 31, 191, 257, 367, 3061, 110503,... | |
| −35 | 11, 13, 79, 127, 503, 617, 709, 857, 1499, 3823, 135623,... | |
| −34 | 3, 294277,... | |
| −33 | 5, 67, 157, 12211,... | |
| −32 | 2* | |
| −31 | 109, 461, 1061, 50777,... | |
| −30 | 2*, 139, 173, 547, 829, 2087, 2719, 3109, 10159, 56543, 80599,... | |
| −29 | 7, 112153, 151153,... | |
| −28 | 3, 19, 373, 419, 491, 1031, 83497,... | |
| −27 | ||
| −26 | 11, 109, 227, 277, 347, 857, 2297, 9043,... | |
| −25 | 3, 7, 23, 29, 59, 1249, 1709, 1823, 1931, 3433, 8863, 43201, 78707,... | |
| −24 | 2*, 7, 11, 19, 2207, 2477, 4951,... | |
| −23 | 11, 13, 67, 109, 331, 587, 24071, 29881, 44053,... | |
| −22 | 3, 5, 13, 43, 79, 101, 107, 227, 353, 7393, 50287,... | |
| −21 | 3, 5, 7, 13, 37, 347, 17597, 59183, 80761, 210599, 394579,... | |
| −20 | 2*, 5, 79, 89, 709, 797, 1163, 6971, 140053, 177967, 393257,... | |
| −19 | 17, 37, 157, 163, 631, 7351, 26183, 30713, 41201, 77951, 476929,... | |
| −18 | 2*, 3, 7, 23, 73, 733, 941, 1097, 1933, 4651, 481147,... | |
| −17 | 7, 17, 23, 47, 967, 6653, 8297, 41221, 113621, 233689, 348259,... | |
| −16 | 3, 5, 7, 23, 37, 89, 149, 173, 251, 307, 317, 30197, 1025393,... | |
| −15 | 3, 7, 29, 1091, 2423, 54449, 67489, 551927,... | |
| −14 | 2*, 7, 53, 503, 1229, 22637, 1091401,... | |
| −13 | 3, 11, 17, 19, 919, 1151, 2791, 9323, 56333, 1199467,... | |
| −12 | 2*, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, 495953,... | |
| −11 | 5, 7, 179, 229, 439, 557, 6113, 223999, 327001,... | |
| −10 | 5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207,... | |
| −9 | 3, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247, 703393,... | |
| −8 | 2* | |
| −7 | 3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653, 1178033,... | |
| −6 | 2*, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, 1313371,... | |
| −5 | 5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, 1856147,... | |
| −4 | 2*, 3 | |
| −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,... | |
| −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,... | |
| 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,... | |
| 3 | 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303,... | |
| 4 | 2 | |
| 5 | 3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279,... | |
| 6 | 2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099,... | |
| 7 | 5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699,... | |
| 8 | 3 | |
| 9 | ||
| 10 | 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343,... | |
| 11 | 17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831,... | |
| 12 | 2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543,... | |
| 13 | 5, 7, 137, 283, 883, 991, 1021, 1193, 3671, 18743, 31751, 101089,... | |
| 14 | 3, 7, 19, 31, 41, 2687, 19697, 59693, 67421, 441697,... | |
| 15 | 3, 43, 73, 487, 2579, 8741, 37441, 89009, 505117, 639833,... | |
| 16 | 2 | |
| 17 | 3, 5, 7, 11, 47, 71, 419, 4799, 35149, 54919, 74509,... | |
| 18 | 2, 25667, 28807, 142031, 157051, 180181, 414269,... | |
| 19 | 19, 31, 47, 59, 61, 107, 337, 1061, 9511, 22051, 209359,... | |
| 20 | 3, 11, 17, 1487, 31013, 48859, 61403, 472709,... | |
| 21 | 3, 11, 17, 43, 271, 156217, 328129,... | |
| 22 | 2, 5, 79, 101, 359, 857, 4463, 9029, 27823,... | |
| 23 | 5, 3181, 61441, 91943, 121949,... | |
| 24 | 3, 5, 19, 53, 71, 653, 661, 10343, 49307, 115597, 152783,... | |
| 25 | ||
| 26 | 7, 43, 347, 12421, 12473, 26717,... | |
| 27 | 3 | |
| 28 | 2, 5, 17, 457, 1423, 115877,... | |
| 29 | 5, 151, 3719, 49211, 77237,... | |
| 30 | 2, 5, 11, 163, 569, 1789, 8447, 72871, 78857, 82883,... | |
| 31 | 7, 17, 31, 5581, 9973, 101111,... | |
| 32 | ||
| 33 | 3, 197, 3581, 6871, 183661,... | |
| 34 | 13, 1493, 5851, 6379, 125101,... | |
| 35 | 313, 1297,... | |
| 36 | 2 | |
| 37 | 13, 71, 181, 251, 463, 521, 7321, 36473, 48157, 87421, 168527,... | |
| 38 | 3, 7, 401, 449, 109037,... | |
| 39 | 349, 631, 4493, 16633, 36341,... | |
| 40 | 2, 5, 7, 19, 23, 29, 541, 751, 1277,... | |
| 41 | 3, 83, 269, 409, 1759, 11731,... | |
| 42 | 2, 1319,... | |
| 43 | 5, 13, 6277, 26777, 27299, 40031, 44773,... | |
| 44 | 5, 31, 167, 100511,... | |
| 45 | 19, 53, 167, 3319, 11257, 34351,... | |
| 46 | 2, 7, 19, 67, 211, 433, 2437, 2719, 19531,... | |
| 47 | 127, 18013, 39623,... | |
| 48 | 19, 269, 349, 383, 1303, 15031,... | |
| 49 | ||
| 50 | 3, 5, 127, 139, 347, 661, 2203, 6521,... |
* 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.
For more information, see.
Algebra factorization of generalized repunit numbers
If b is a perfect power differs from 1, then there is at most one repunit in base-b. If n is a prime power, 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 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, b = 1024, and b = −1 or 0. Another special situation is b = −4k4, with k positive integer, which has the aurifeuillean factorization, for example, b = −4, and b = −64, −324, −1024, −2500, −5184,..., 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.The generalized repunit conjecture
A conjecture related to the generalized repunit primes:For any integer, which satisfies the conditions:
- .
- is not a perfect power.
- is not in the form.
for prime, the prime numbers will be distributed near the best fit line
where limit,
and there are about
base-b repunit primes less than N.
- is the base of natural logarithm.
- is Euler–Mascheroni constant.
- is the logarithm in base
- is the th generalized repunit prime in baseb
- is a data fit constant which varies with.
- if, if.
- is the largest natural number such that is a th power.
- The number of prime numbers of the form less than or equal to is about.
- The expected number of prime numbers of the form with prime between and is about.
- The probability that number of the form is prime is about.
History
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 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 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 the repunits to base 2, 3, 5, 6, 7, 10, 11, and 12.
Demlo numbers
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 and must add up to a repdigit number, and the middle part may contain any additional number of this repeated digit.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, 1, 121, 12321,..., 12345678987654321, 1234567900987654321, 123456790120987654321,...,, although one can check these are not Demlo numbers for p = 10, 19, 28,...