Repunit
No. of known terms  9 

Conjectured no. of terms  Infinite 
First terms  11, 1111111111111111111, 11111111111111111111111 
Largest known term  (10^{270343}1)/9 
OEIS index 

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.
Contents
 1 Definition
 2 Properties
 3 Factorization of decimal repunits
 4 Repunit primes
 4.1 Decimal repunit primes
 4.2 Base 2 repunit primes
 4.3 Base 3 repunit primes
 4.4 Base 4 repunit primes
 4.5 Base 5 repunit primes
 4.6 Base 6 repunit primes
 4.7 Base 7 repunit primes
 4.8 Base 8 repunit primes
 4.9 Base 9 repunit primes
 4.10 Base 11 repunit primes
 4.11 Base 12 repunit primes
 4.12 Base 20 repunit primes
 4.13 Bases such that is prime for prime
 4.14 List of repunit primes base
 4.15 Algebra factorization of generalized repunit numbers
 4.16 The generalized repunit conjecture
 5 History
 6 Demlo numbers
 7 See also
 8 Footnotes
 9 References
 10 External links
Definition[edit]
The baseb repunits are defined as (this b can be either positive or negative)
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
In particular, the decimal (base10) repunits that are often referred to as simply repunits are defined as
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
Similarly, the repunits base 2 are defined as
Thus, the number R_{n}^{(2)} consists of n copies of the digit 1 in base 2 representation. In fact, the base2 repunits are the wellknown Mersenne numbers M_{n} = 2^{n} − 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[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,
 R_{35}^{(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 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 pigeonhole 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 R_{1}^{(b)},...,R_{n}^{(b)}. Because there are n repunits but only n1 nonzero 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_{ji}^{(b)} x b^{i} . Since n divides the lefthand side it also divides the righthand side and since n and b are relatively prime n must divide R_{ji}^{(b)}.
 The Feit–Thompson conjecture is that R_{q}^{(p)} never divides R_{p}^{(q)} for two distinct primes p and q.
 Using the Euclidean Algorithm for repunits definition: R_{1}^{(b)} = 1; R_{n}^{(b)} = R_{n1}^{(b)} x b + 1, any consecutive repunits R_{n1}^{(b)} and R_{n}^{(b)} are relatively prime in any base b for any n.
 If m and n are relatively prime, R_{m}^{(b)} and R_{n}^{(b)} are relatively prime in any base b for any m and n. The Euclidean Algorithm is based on gcd(m, n) = gcd(m  n, n) for m > n. Similarly, using R_{m}^{(b)}  R_{n}^{(b)} × b^{mn} = R_{mn}^{(b)}, it can be easily shown that gcd(R_{m}^{(b)}, R_{n}^{(b)}) = gcd(R_{mn}^{(b)}, R_{n}^{(b)}) for m > n. Therefore if gcd(m, n) = 1, then gcd(R_{m}^{(b)}, R_{n}^{(b)}) = R_{1}^{(b)} = 1.
 The remainder of R_{n}^{(10)} modulo 3 is equal to the remainder of n modulo 3. Using 10^{a} ≡ 1 (mod 3) for any a ≥ 0,
n ≡ 0 (mod 3) ⇔ R_{n}^{(10)} ≡ 0 (mod 3) ⇔ R_{n}^{(10)} ≡ 0 (mod R_{3}^{(10)}),
n ≡ 1 (mod 3) ⇔ R_{n}^{(10)} ≡ 1 (mod 3) ⇔ R_{n}^{(10)} ≡ R_{1}^{(10)} ≡ 1 (mod R_{3}^{(10)}),
n ≡ 2 (mod 3) ⇔ R_{n}^{(10)} ≡ 2 (mod 3) ⇔ R_{n}^{(10)} ≡ R_{2}^{(10)} ≡ 11 (mod R_{3}^{(10)}).
Therefore, 3  n ⇔ 3  R_{n}^{(10)} ⇔ R_{3}^{(10)}  R_{n}^{(10)}.  The remainder of R_{n}^{(10)} modulo 9 is equal to the remainder of n modulo 9. Using 10^{a} ≡ 1 (mod 9) for any a ≥ 0,
n ≡ r (mod 9) ⇔ R_{n}^{(10)} ≡ r (mod 9) ⇔ R_{n}^{(10)} ≡ R_{r}^{(10)} (mod R_{9}^{(10)}),
for 0 ≤ r < 9.
Therefore, 9  n ⇔ 9  R_{n}^{(10)} ⇔ R_{9}^{(10)}  R_{n}^{(10)}.
Factorization of decimal repunits[edit]
(Prime factors colored red means "new factors", i. e. the prime factor divides R_{n} but not divides R_{k} for all k < n) (sequence A102380 in the OEIS)^{[2]}



Smallest prime factor of R_{n} for n > 1 are
 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[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 R_{n}^{(b)} is divisible by R_{a}^{(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 R_{9} is divisible by R_{3}—in fact, 111111111 = 111 · 1001001. The corresponding cyclotomic polynomials and are and , 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, R_{3} = 111 = 3 · 37 is not prime. Except for this case of R_{3}, p can only divide R_{n} for prime n if p = 2kn + 1 for some k.
Decimal repunit primes[edit]
R_{n} is prime for n = 2, 19, 23, 317, 1031, ... (sequence A004023 in OEIS). R_{49081} and R_{86453} are probably prime. On April 3, 2007 Harvey Dubner (who also found R_{49081}) announced that R_{109297} is a probable prime.^{[3]} He later announced there are no others from R_{86453} to R_{200000}.^{[4]} On July 15, 2007 Maksym Voznyy announced R_{270343} 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[edit]
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 the OEIS),
corresponding to of
Base 4 repunit primes[edit]
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[edit]
The first few base 5 repunit primes are
 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531, 35032461608120426773093239582247903282006548546912894293926707097244777067146515037165954709053039550781, 815663058499815565838786763657068444462645532258620818469829556933715405574685778402862015856733535201783524826169013977050781 (sequence A086122 in the OEIS),
corresponding to of
Base 6 repunit primes[edit]
The first few base 6 repunit primes are
 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371, 133733063818254349335501779590081460423013416258060407531857720755181857441961908284738707408499507 (sequence A165210 in the OEIS),
corresponding to of
Base 7 repunit primes[edit]
The first few base 7 repunit primes are
 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457,
138502212710103408700774381033135503926663324993317631729227790657325163310341833227775945426052637092067324133850503035623601
corresponding to of
Base 8 repunit primes[edit]
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[edit]
There are no base 9 repunit primes. , both and are even and greater than 4.
Base 11 repunit primes[edit]
The first few base 11 repunit primes are
 50544702849929377, 6115909044841454629, 1051153199500053598403188407217590190707671147285551702341089650185945215953, 567000232521795739625828281267171344486805385881217575081149660163046217465544573355710592079769932651989153833612198334843467861091902034340949
corresponding to of
Base 12 repunit primes[edit]
The first few base 12 repunit primes are
 13, 157, 22621, 29043636306420266077, 43570062353753446053455610056679740005056966111842089407838902783209959981593077811330507328327968191581, 388475052482842970801320278964160171426121951256610654799120070705613530182445862582590623785872890159937874339918941
corresponding to of
Base 20 repunit primes[edit]
The first few base 20 repunit primes are
 421, 10778947368421, 689852631578947368421
corresponding to of
Bases such that is prime for prime [edit]
Smallest base such that is prime (where is the th prime) are
 2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, 13, 136, 220, 162, 35, 10, 218, 19, 26, 39, 12, 22, 67, 120, 195, 48, 54, 463, 38, 41, 17, 808, 404, 46, 76, 793, 38, 28, 215, 37, 236, 59, 15, 514, 260, 498, 6, 2, 95, 3, ... (sequence A066180 in the OEIS)
Smallest base such that is prime (where is the th prime) are
 3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, 159, 10, 16, 209, 2, 16, 23, 273, 2, 460, 22, 3, 36, 28, 329, 43, 69, 86, 271, 396, 28, 83, 302, 209, 11, 300, 159, 79, 31, 331, 52, 176, 3, 28, 217, 14, 410, 252, 718, 164, ... (sequence A103795 in the OEIS)
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, ... 
List of repunit primes base [edit]
Smallest prime such that is prime are (start with , 0 if no such exists)
 3, 3, 0, 3, 3, 5, 3, 0, 19, 17, 3, 5, 3, 3, 0, 3, 25667, 19, 3, 3, 5, 5, 3, 0, 7, 3, 5, 5, 5, 7, 0, 3, 13, 313, 0, 13, 3, 349, 5, 3, 1319, 5, 5, 19, 7, 127, 19, 0, 3, 4229, 103, 11, 3, 17, 7, 3, 41, 3, 7, 7, 3, 5, 0, 19, 3, 19, 5, 3, 29, 3, 7, 5, 5, 3, 41, 3, 3, 5, 3, 0, 23, 5, 17, 5, 11, 7, 61, 3, 3, 4421, 439, 7, 5, 7, 3343, 17, 13, 3, 0, ... (sequence A128164 in the OEIS)
Smallest prime such that is prime are (start with , 0 if no such exists, question mark if this term is currently unknown)
 3, 3, 3, 5, 3, 3, 0, 3, 5, 5, 5, 3, 7, 3, 3, 7, 3, 17, 5, 3, 3, 11, 7, 3, 11, 0, 3, 7, 139, 109, 0, 5, 3, 11, 31, 5, 5, 3, 53, 17, 3, 5, 7, 103, 7, 5, 5, 7, 1153, 3, 7, 21943, 7, 3, 37, 53, 3, 17, 3, 7, 11, 3, 0, 19, 7, 3, 757, 11, 3, 5, 3, 7, 13, 5, 3, 37, 3, 3, 5, 3, 293, 19, 7, 167, 7, 7, 709, 13, 3, 3, 37, 89, 71, 43, 37, ?, 19, 7, 3, ... (sequence A084742 in the OEIS)
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, ...  
−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, 127609, ...  A231604 
−41  17, 691, 113749, ...  
−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, ...  
−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, ...  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, ..., 77232917, ...  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, 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, ...  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, ...  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, ...  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.
For more information, see.^{[7]}^{[8]}^{[9]}^{[10]}
Algebra factorization of generalized repunit numbers[edit]
If b is a perfect power (can be written as m^{n}, 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 R_{2}. 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 R_{2} 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 R_{2} 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 = −4k^{4}, with k positive integer, which has the aurifeuillean factorization, for example, b = −4 (with k = 1, then R_{2} and R_{3} 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 −4k^{4} with k positive integer, then there are infinity many base b repunit primes.
The generalized repunit conjecture[edit]
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 )
For any integer , which satisfies the conditions:
 .
 is not a perfect power. (since when is a perfect th power, it can be shown that there is at most one value such that is prime, and this value is itself or a root of )
 is not in the form . (if so, then the number has aurifeuillean factorization)
has generalized repunit primes of the form
for prime , the prime numbers will be distributed near the best fit line
where limit ,
and there are about
base repunit primes less than .
 is the base of natural logarithm.
 is Euler–Mascheroni constant.
 is the logarithm in base
 is the th generalized repunit prime in base (with prime )
 is a data fit constant which varies with .
 if , if .
 is the largest natural number such that is a th power.
We also have the following 3 properties:
 The number of prime numbers of the form (with prime ) 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 (for prime ) is about .
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.^{[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 R_{16} and many larger ones. By 1880, even R_{17} to R_{36} 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 R_{19} to be prime in 1916^{[14]} and Lehmer and Kraitchik independently found R_{23} 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. R_{317} was found to be a probable prime circa 1966 and was proved prime eleven years later, when R_{1031} 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 sidedevelopment 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]
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, ...
See also[edit]
 All one polynomial  Another generalization
 Goormaghtigh conjecture
 Repeating decimal
 Repdigit
 Wagstaff prime  can be thought of as repunit primes with negative base
Footnotes[edit]
Notes[edit]
 ^ Albert H. Beiler coined the term “repnunit number” as follows:
A number which consists of a repeated of a single digit is sometimes called a monodigit number, and for convenience the author has used the term “repunit number”(repeated unit) to represent monodigit numbers consisting solely of the digit 1.^{[1]}
References[edit]
 ^ Beiler 2013, pp. 83
 ^ 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 160
 ^ Repunit primes in base −160 to −2
 ^ Repunit primes in base −200 to −2
 ^ Deriving the Wagstaff Mersenne Conjecture
 ^ Generalized Repunit Conjecture
 ^ ^{a} ^{b} Dickson & Cresse 1999, pp. 164–167
 ^ Francis 1988, pp. 240–246
 ^ Kaprekar 1938, Gunjikar and Kaprekar 1939
 ^ Weisstein, Eric W. "Demlo Number". MathWorld.
References[edit]
 Beiler, Albert H. (2013) [1964], Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover Recreational Math (2nd Revised ed.), New York: Dover Publications, ISBN 9780486210964
 Dickson, Leonard Eugene; Cresse, G.H. (19990424), History of the Theory of Numbers, AMS Chelsea Publishing, Volume I (2nd Reprinted ed.), Providence, Rhode Island: American Mathematical Society, ISBN 9780821819340
 Francis, Richard L. (1988), "Mathematical Haystacks: Another Look at Repunit Numbers", The College Mathematics Journal, 19 (3): 240–246
 Gunjikar, K. R.; Kaprekar, D. R. (1939), "Theory of Demlo numbers" (PDF), Journal of the University of Bombay, VIII (3): 3–9
 Kaprekar, D. R. (1938), "On Wonderful Demlo numbers", The Mathematics Student, 6: 68
 Kaprekar, D. R. (1938), "Demlo numbers", J. Phys. Sci. Univ. Bombay, VII (3)
 Kaprekar, D. R. (1948), Demlo numbers, Devlali, India: Khareswada
 Ribenboim, Paulo (19960202), The New Book of Prime Number Records, Computers and Medicine (3rd ed.), New York: Springer, ISBN 9780387944579
 Yates, Samuel (1982), Repunits and repetends, FL: Delray Beach, ISBN 9780960865208
External links[edit]
 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.
 Smallest odd prime p such that (b^p1)/(b1) and (b^p+1)/(b+1) is prime for bases 2<=b<=1024
 Factorization of repunit numbers
 Generalized repunit primes in base 50 to 50