Repunit

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Repunit prime
Number of known terms 9
Conjectured number of terms Infinite
First terms 11, 1111111111111111111, 11111111111111111111111
Largest known term (10270343-1)/9
OEIS index A004022

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.[1]

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

Definition[edit]

The base-b repunits are defined as (this b can be either positive or negative)

R_n^{(b)}={b^n-1\over{b-1}}\qquad\mbox{for }|b|\ge2, n\ge1.

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

R_1^{(b)}={b-1\over{b-1}}= 1 \qquad \text{and} \qquad R_2^{(b)}={b^2-1\over{b-1}}= b+1\qquad\text{for}\ |b|\ge2.

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

R_n=R_n^{(10)}={10^n-1\over{10-1}}={10^n-1\over9}\qquad\mbox{for }n\ge1.

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

1, 11, 111, 1111, 11111, 111111, ... (sequence A002275 in OEIS).

Similarly, the repunits base 2 are defined as

R_n^{(2)}={2^n-1\over{2-1}}={2^n-1}\qquad\mbox{for }n\ge1.

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

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, ... (sequence A000225 in OEIS)

Properties[edit]

  • Any repunit in any base having a composite number of digits is necessarily composite. Only repunits (in any base) having a prime number of digits might be prime. This is a necessary but not sufficient condition. For example,
    R35(b) = 11111111111111111111111111111111111 = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001,
since 35 = 7 × 5 = 5 × 7. This repunit factorization does not depend on the base b in which the repunit is expressed.
  • Any positive multiple of the repunit Rn(b) contains at least n nonzero digits in base b.
  • The only known numbers that are repunits with at least 3 digits in more than one base simultaneously are 31 (111 in base 5, 11111 in base 2) and 8191 (111 in base 90, 1111111111111 in base 2). The Goormaghtigh conjecture says there are only these two cases.
  • Using the pigeon-hole principle it can be easily shown that for each n and b such that n and b are relatively prime there exists a repunit in base b that is a multiple of n. To see this consider repunits R1(b),...,Rn(b). Assume none of the Rk(b) is divisible by n. Because there are n repunits but only n-1 non-zero residues modulo n there exist two repunits Ri(b) and Rj(b) with 1≤i<jn such that Ri(b) and Rj(b) have the same residue modulo n. It follows that Rj(b) - Ri(b) has residue 0 modulo n, i.e. is divisible by n. Rj(b) - Ri(b) consists of j - i ones followed by i zeroes. Thus, Rj(b) - Ri(b) = Rj-i(b) x bi . Since n divides the left-hand side it also divides the right-hand side and since n and b are relative prime n must divide Rj-i(b) contradicting the original assumption.
  • The Feit–Thompson conjecture is that Rq(p) never divides Rp(q) for two distinct primes p and q.

Factorization of decimal repunits[edit]

(Prime factors colored red means "new factors", the prime factor divides Rn but not divides Rk for all k < n) (sequence A102380 in OEIS)

R1 = 1
R2 = 11
R3 = 3 · 37
R4 = 11 · 101
R5 = 41 · 271
R6 = 3 · 7 · 11 · 13 · 37
R7 = 239 · 4649
R8 = 11 · 73 · 101 · 137
R9 = 32 · 37 · 333667
R10 = 11 · 41 · 271 · 9091
R11 = 21649 · 513239
R12 = 3 · 7 · 11 · 13 · 37 · 101 · 9901
R13 = 53 · 79 · 265371653
R14 = 11 · 239 · 4649 · 909091
R15 = 3 · 31 · 37 · 41 · 271 · 2906161
R16 = 11 · 17 · 73 · 101 · 137 · 5882353
R17 = 2071723 · 5363222357
R18 = 32 · 7 · 11 · 13 · 19 · 37 · 52579 · 333667
R19 = 1111111111111111111
R20 = 11 · 41 · 101 · 271 · 3541 · 9091 · 27961
R21 = 3 · 37 · 43 · 239 · 1933 · 4649 · 10838689
R22 = 112 · 23 · 4093 · 8779 · 21649 · 513239
R23 = 11111111111111111111111
R24 = 3 · 7 · 11 · 13 · 37 · 73 · 101 · 137 · 9901 · 99990001
R25 = 41 · 271 · 21401 · 25601 · 182521213001
R26 = 11 · 53 · 79 · 859 · 265371653 · 1058313049
R27 = 33 · 37 · 757 · 333667 · 440334654777631
R28 = 11 · 29 · 101 · 239 · 281 · 4649 · 909091 · 121499449
R29 = 3191 · 16763 · 43037 · 62003 · 77843839397
R30 = 3 · 7 · 11 · 13 · 31 · 37 · 41 · 211 · 241 · 271 · 2161 · 9091 · 2906161

Smallest prime factor of Rn are

1, 11, 3, 11, 41, 3, 239, 11, 3, 11, 21649, 3, 53, 11, 3, 11, 2071723, 3, 1111111111111111111, 11, 3, 11, 11111111111111111111111, 3, 41, 11, 3, 11, 3191, 3, 2791, 11, 3, 11, 41, 3, 2028119, 11, 3, 11, 83, 3, 173, 11, 3, 11, 35121409, 3, 239, 11, ... (sequence A067063 in OEIS)

Repunit primes[edit]

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):

R_n^{(b)}=\frac{1}{b-1}\prod_{d|n}\Phi_d(b)

where \Phi_d(x) is the d^\mathrm{th} cyclotomic polynomial and d ranges over the divisors of n. For p prime, \Phi_p(x)=\sum_{i=0}^{p-1}x^i, 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 \Phi_3(x) and \Phi_9(x) are x^2+x+1 and x^6+x^3+1 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[edit]

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.[2] He later announced there are no others from R86453 to R200000.[3] On July 15, 2007 Maksym Voznyy announced R270343 to be probably prime,[4] 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[5] 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.

Base 2 repunit primes[edit]

Main article: Mersenne prime

Base 2 repunit primes are called Mersenne primes.

Base 3 repunit primes[edit]

The first few base 3 repunit primes are

13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 (sequence A076481 in OEIS)

corresponding to n of

3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, ... (sequence A028491 in OEIS).

Base 4 repunit primes[edit]

The only base 4 repunit prime is 5 (11_4). 4^n-1=\left(2^n+1\right)\left(2^n-1\right), and 3 always divides 2^n+1 when n is odd and 2^n-1 when n is even. For n greater than 2, both 2^n+1 and 2^n-1 are greater than 3, so removing the factor of 3 still leaves two factors greater than 1, so the number cannot be prime.

Base 5 repunit primes[edit]

The first few base 5 repunit primes are

31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531, 35032461608120426773093239582247903282006548546912894293926707097244777067146515037165954709053039550781, 815663058499815565838786763657068444462645532258620818469829556933715405574685778402862015856733535201783524826169013977050781 (sequence A086122 in OEIS)

corresponding to n of

3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, ... (sequence A004061 in OEIS).

Base 6 repunit primes[edit]

The first few base 6 repunit primes are

7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371, 133733063818254349335501779590081460423013416258060407531857720755181857441961908284738707408499507 (sequence A165210 in OEIS)

corresponding to n of

2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, ... (sequence A004062 in OEIS)

Base 7 repunit primes[edit]

The first few base 7 repunit primes are

2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457,
138502212710103408700774381033135503926663324993317631729227790657325163310341833227775945426052637092067324133850503035623601

corresponding to n of

5, 13, 131, 149, 1699, ... (sequence A004063 in OEIS)

Base 8 and 9 repunit primes[edit]

The only base 8 or base 9 repunit prime is 73 (111_8). 8^n-1=\left(4^n+2^n+1\right)\left(2^n-1\right), and 7 divides 4^n+2^n+1 when n is not divisible by 3 and 2^n-1 when n is a multiple of 3. 9^n-1=\left(3^n+1\right)\left(3^n-1\right), and 2 always divides both 3^n+1 and 3^n-1.

Base 12 repunit primes[edit]

The first few base 12 repunit primes are

13, 157, 22621, 29043636306420266077, 435700623537534460534556100566709740005056966111842089407838902783209959981593077811330507328327968191581, 388475052482842970801320278964160171426121951256610654799120070705613530182445862582590623785872890159937874339918941

corresponding to n of

2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, ... (sequence A004064 in OEIS)

Base 20 repunit primes[edit]

The first few base 20 repunit primes are

421, 10778947368421, 689852631578947368421

corresponding to n of

3, 11, 17, 1487, ... (sequence A127995 in OEIS)

Bases b such that R_p(b) is prime for prime p[edit]

(For the smallest base b such that R_p(b) is prime for prime p, see OEISA066180 (positive bases), OEISA103795 (negative bases))

p bases b such that R_p(b) 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, ... 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, ...

List of repunit primes base b[edit]

(For the smallest n such that R_n(b) is prime, see OEISA084740 (positive bases), OEISA128164 (positive bases, n = 2 not allowed), OEISA084742 (negative bases, n = 2 not allowed))

b numbers n such that R_n(b) is prime (some large terms are only PRP, these n are checked up to 100000) OEIS sequence
−50 1153, 26903, 56597, ...
−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, ...
−44 2*, 7, 41233, ...
−43 5, 7, 19, 251, 277, 383, 503, 3019, 4517, 9967, 29573, ... A231865
−42 2*, 3, 709, 1637, 17911, ... A231604
−41 17, 691, ...
−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, ... A229524
−37 5, 7, 2707, ...
−36 31, 191, 257, 367, 3061, ... A229145
−35 11, 13, 79, 127, 503, 617, 709, 857, 1499, 3823, ... A185240
−34 3, ...
−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, ...
−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, ... 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, ... A057179
−12 2*, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... 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, ... A057173
−6 2*, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, ... A057172
−5 5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, ... 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, ... A000043
3 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, ... A028491
4 2 (no others)
5 3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, ... 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, ... 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, ... A204940
24 3, 5, 19, 53, 71, 653, 661, 10343, 49307, ... A127998
25 (none)
26 7, 43, 347, 12421, 12473, 26717, ... A127999
27 3 (no others)
28 2, 5, 17, 457, 1423, ... 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, ... A209120
34 13, 1493, 5851, 6379, ... A185073
35 313, 1297, ...
36 2 (no others)
37 13, 71, 181, 251, 463, 521, 7321, 36473, 48157, 87421, ... A128003
38 3, 7, 401, 449, ... 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, ...
45 19, 53, 167, 3319, 11257, 34351, ... A242797
46 2, 7, 19, 67, 211, 433, 2437, 2719, 19531, ... A243279
47 127, 18013, 39623, ...
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.

For more information, see Repunit primes in base −50 to 50, Repunit primes in base 2 to 150, Repunit primes in base −150 to −2, and Repunit primes in base −200 to −2.

Algebra factorization of repunit numbers[edit]

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 letter 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.

History[edit]

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.[6]

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[7] 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[8] 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.

Demlo numbers[edit]

The Demlo numbers[9] 1, 121, 12321, 1234321, ..., 12345678987654321, 1234567900987654321, 123456790120987654321, ..., (sequence A002477 in OEIS) were defined by D. R. Kaprekar as the squares of the repunits, resolving the uncertainty how to continue beyond the highest digit (9), and named after Demlo railway station 30 miles from Bombay on the then G.I.P. Railway, where he thought of investigating them.

See also[edit]

Notes[edit]

  1. ^ Beiler, Albert (2013). Recreations in the Theory of Numbers: The Queen of Mathematics Entertains (2 ed.). New York: Dover Publications. p. 83. ISBN 978-0-486-21096-4. 
  2. ^ Harvey Dubner, New Repunit R(109297)
  3. ^ Harvey Dubner, Repunit search limit
  4. ^ Maksym Voznyy, New PRP Repunit R(270343)
  5. ^ Chris Caldwell, "The Prime Glossary: repunit" at The Prime Pages.
  6. ^ Dickson, Leonard Eugene and Cresse, G.H.; History of the Theory of Numbers; pp. 164-167 ISBN 0-8218-1934-8
  7. ^ Dickson and Cresse, pp. 164-167
  8. ^ Francis, Richard L.; "Mathematical Haystacks: Another Look at Repunit Numbers" in The College Mathematics Journal, Vol. 19, No. 3. (May, 1988), pp. 240-246.
  9. ^ Weisstein, Eric W., "Demlo Number", MathWorld.

External links[edit]

Web sites[edit]

Books[edit]