Repunit

Repunit prime
No. of known terms	9
Conjectured no. 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

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

R_{n}^{(b)}\equiv {b^{n}-1 \over {b-1}}\qquad {\mbox{for }}|b|\geq 2,n\geq 1.

Thus, the number R_n^(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|\geq 2.

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

R_{n}\equiv R_{n}^{(10)}={10^{n}-1 \over {10-1}}={10^{n}-1 \over 9}\qquad {\mbox{for }}n\geq 1.

Thus, the number R_n = R_n⁽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 the 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\geq 1.

Thus, the number R_n⁽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 M_n = 2ⁿ − 1, they start with

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

Properties

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) = 35 = 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 R_n^(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),...,R_n^(b). Assume none of the R_k^(b) is divisible by n. Because there are n repunits but only n-1 non-zero residues modulo n there exist two repunits R_i^(b) and R_j^(b) with 1≤i<j≤n such that R_i^(b) and R_j^(b) have the same residue modulo n. It follows that R_j^(b) - R_i^(b) has residue 0 modulo n, i.e. is divisible by n. R_j^(b) - R_i^(b) consists of j - i ones followed by i zeroes. Thus, R_j^(b) - R_i^(b) = R_j-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 R_j-i^(b) contradicting the original assumption.
The Feit–Thompson conjecture is that R_q^(p) never divides R_p^(q) for two distinct primes p and q.

Factorization of decimal repunits

(Prime factors colored red means "new factors", the prime factor divides R_n but not divides R_k for all k < n) (sequence A102380 in the OEIS)^[2]

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 =	19
R₂₀ =	11 · 41 · 101 · 271 · 3541 · 9091 · 27961

R₂₁ =	3 · 37 · 43 · 239 · 1933 · 4649 · 10838689
R₂₂ =	11² · 23 · 4093 · 8779 · 21649 · 513239
R₂₃ =	23
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 R_n 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 the OEIS)

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 R_n^(b) is divisible by R_a^(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 R_n 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 R_n for prime n if p = 2kn + 1 for some k.

Decimal repunit primes

R_n 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 R_2500000 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.

Base 2 repunit primes

Base 2 repunit primes are called Mersenne primes.

Base 3 repunit primes

The first few base 3 repunit primes are

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

corresponding to $n$ of

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

Base 4 repunit primes

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. Therefore, the number cannot be prime.

Base 5 repunit primes

The first few base 5 repunit primes are

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

corresponding to $n$ of

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

Base 6 repunit primes

The first few base 6 repunit primes are

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

corresponding to $n$ of

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

Base 7 repunit primes

The first few base 7 repunit primes are

2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457,
138502212710103408700774381033135503926663324993317631729227790657325163310341833227775945426052637092067324133850503035623601,

corresponding to $n$ of

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

Base 8 repunit primes

The only base 8 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.

Base 9 repunit primes

There are no base 9 repunit primes. $9^{n}-1=\left(3^{n}+1\right)\left(3^{n}-1\right)$ , and 2 always divides both $3^{n}+1$ and $3^{n}-1$ .

Base 12 repunit primes

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 the OEIS).

Base 20 repunit primes

The first few base 20 repunit primes are

421, 10778947368421, 689852631578947368421,

corresponding to $n$ of

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

Bases $b$ such that $R_{p}(b)$ is prime for prime $p$

(For the base $b$ such that $R_{p}(b)$ is prime for prime $p$ , see OEIS: A066180 (smallest positive base), OEIS: A103795 (largest negative base))

$p$	bases $b$ such that $R_{p}(b)$ 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, ...

List of repunit primes base $b$

(For the smallest $n$ such that $R_{n}(b)$ is prime, see OEIS: A084740 (positive bases), OEIS: A128164 (positive bases, $n$ = 2 not allowed), OEIS: A084742 (negative bases, $n$ = 2 not allowed))

$b$	numbers $n$ such that $R_{n}(b)$ is prime (some large terms are only probable primes, 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, 1199467, ...	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, ..., 74207281, ...	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, ...	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.

For more information, see.^[7]^[8]^[9]^[10]

Algebra factorization of repunit numbers

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 p^r, with p prime, r integer, p, r >0), then all repunit in base b are not prime aside from R_p and R₂. R_p 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.

The generalized repunit conjecture

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 $b$ )

For any integer $b$ , which satisfies the conditions:

$|b|>1$ .
$b$ is not a perfect power. (since when $b$ is a perfect $r$ th power, it can be shown that there is at most one $n$ value such that ${\frac {b^{n}-1}{b-1}}$ is prime, and this $n$ value is $r$ itself or a root of $r$ )
$b$ is not in the form $-4k^{4}$ . (if so, then the number has aurifeuillean factorization)

has generalized repunit primes of the form

R_{p}(b)={\frac {b^{p}-1}{b-1}}

for prime $p$ , the prime numbers will be distributed near the best fit line

Y=G\cdot \log _{|b|}\left(\log _{|b|}\left(R_{(b)}(n)\right)\right)+C,

where limit $n\rightarrow \infty$ , $G={\frac {1}{e^{\gamma }}}=0.561459483566...$

and there are about

\left(\log _{e}(N)+m\cdot \log _{e}(2)\cdot \log _{e}{\big (}\log _{e}(N){\big )}+{\frac {1}{\sqrt {N}}}-\delta \right)\cdot {\frac {e^{\gamma }}{\log _{e}(|b|)}}

base $b$ repunit primes less than $N$ .

$e$ is the base of natural logarithm.
$\gamma$ is Euler–Mascheroni constant.
$\log _{|b|}$ is the logarithm in base $|b|$
$R_{(b)}(n)$ is the $n$ th generalized repunit prime in base $b$ (with prime $p$ )
$C$ is a data fit constant which varies with $b$ .
$\delta =1$ if $b>0$ , $\delta =1.6$ if $b<0$ .
$m$ is the largest natural number such that $-b$ is a $2^{m-1}$ th power.

We also have the following 3 properties:

The number of prime numbers of the form ${\frac {b^{n}-1}{b-1}}$ (with prime $p$ ) less than or equal to $n$ is about $e^{\gamma }\cdot \log _{|b|}{\big (}\log _{|b|}(n){\big )}$ .
The expected number of prime numbers of the form ${\frac {b^{n}-1}{b-1}}$ with prime $p$ between $n$ and $|b|\cdot n$ is about $e^{\gamma }$ .
The probability that number of the form ${\frac {b^{n}-1}{b-1}}$ is prime (for prime $p$ ) is about ${\frac {e^{\gamma }}{p\cdot \log _{e}(|b|)}}$ .

History

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^[14] 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^[15] 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

The Demlo numbers^[16] 1, 121, 12321, 1234321, ..., 12345678987654321, 1234567900987654321, 123456790120987654321, ..., (sequence A002477 in the 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.

Notes

^ 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.
^ For more information, see Factorization of repunit numbers.
^ Harvey Dubner, New Repunit R(109297)
^ Harvey Dubner, Repunit search limit
^ Maksym Voznyy, New PRP Repunit R(270343)
^ Chris Caldwell, "The Prime Glossary: repunit" at The Prime Pages.
^ Repunit primes in base −50 to 50
^ Repunit primes in base 2 to 152
^ Repunit primes in base −151 to −2
^ Repunit primes in base −200 to −2
^ Deriving the Wagstaff Mersenne Conjecture
^ Generalized Repunit Conjecture
^ Dickson, Leonard Eugene and Cresse, G.H.; History of the Theory of Numbers; pp. 164-167 ISBN 0-8218-1934-8
^ Dickson and Cresse, pp. 164-167
^ Francis, Richard L.; "Mathematical Haystacks: Another Look at Repunit Numbers" in The College Mathematics Journal, Vol. 19, No. 3. (May, 1988), pp. 240-246.
^ Weisstein, Eric W. "Demlo Number". MathWorld.

External links

Weisstein, Eric W. "Repunit". MathWorld.
The main tables of the Cunningham project.
Repunit at The Prime Pages by Chris Caldwell.
Repunits and their prime factors at World!Of Numbers.
Prime generalized repunits of at least 1000 decimal digits by Andy Steward
Repunit Primes Project Giovanni Di Maria's repunit primes page.
Factorization of repunit numbers
Generalized repunit primes in base -50 to 50

[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] For more information, see Factorization of repunit numbers.

[3] Harvey Dubner, New Repunit R(109297)

[4] Harvey Dubner, Repunit search limit

[5] Maksym Voznyy, New PRP Repunit R(270343)

[6] Chris Caldwell, "The Prime Glossary: repunit" at The Prime Pages.

[7] Repunit primes in base −50 to 50

[8] Repunit primes in base 2 to 152

[9] Repunit primes in base −151 to −2

[10] Repunit primes in base −200 to −2

[11] Deriving the Wagstaff Mersenne Conjecture

[12] Generalized Repunit Conjecture

[13] Dickson, Leonard Eugene and Cresse, G.H.; History of the Theory of Numbers; pp. 164-167 ISBN 0-8218-1934-8

[14] Dickson and Cresse, pp. 164-167

[15] Francis, Richard L.; "Mathematical Haystacks: Another Look at Repunit Numbers" in The College Mathematics Journal, Vol. 19, No. 3. (May, 1988), pp. 240-246.

[16] Weisstein, Eric W. "Demlo Number". MathWorld.

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Repunit

Definition

Properties

Factorization of decimal repunits