# Repunit

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — the simplest form 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. In binary, these are the widely known Mersenne primes.

## Definition

The base-b repunits are defined as

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

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, ... (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.

## 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) = 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

 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

## Repunit primes

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

It is easy to show that if n is divisible by a, then Rn(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

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

Main article: Mersenne prime

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 OEIS),

corresponding to $n$ of

3, 7, 13, 71, 103, ... (sequence A028491 in 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, so the number cannot be prime.

### Base 5 repunit primes

The first few base 5 repunit primes are

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

corresponding to $n$ of

3, 7, 11, 13, 47, ... (sequence A004061 in OEIS).

### Base 6 repunit primes

The first few base 6 repunit primes are

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

corresponding to $n$ of

2, 3, 7, 29, 71, ... (sequence A004062 in 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, ... (sequence A004063 in OEIS)

### Base 8 and 9 repunit primes

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

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, ... (sequence A004064 in OEIS)

### Base 20 repunit primes

The only known vigesimal (base 20) repunit primes or probable primes are for $n$ of

3, 11, 17, 1487, 31013, 48859, 61403 (sequence A127995 in OEIS)

The first three of these in decimal are

421, 10778947368421 and 689852631578947368421

### The smallest repunit prime (p>2) of any natural number base b

The list is about all bases up to 300. (sequence A128164 in OEIS)

 Base +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 0+ 3 3 3 None 3 3 5 3 None 19 17 3 5 3 3 None 3 25667 19 3 20+ 3 5 5 3 None 7 3 5 5 5 7 None 3 13 313 None 13 3 349 5 40+ 3 1319 5 5 19 7 127 19 None 3 4229 103 11 3 17 7 3 41 3 7 60+ 7 3 5 None 19 3 19 5 3 29 3 7 5 5 3 41 3 3 5 3 80+ None 23 5 17 5 11 7 61 3 3 4421 439 7 5 7 3343 17 13 3 None 100+ 3 59 19 97 3 149 17 449 17 3 3 79 23 29 7 59 3 5 3 5 120+ None 5 43 599 None 7 5 7 5 37 3 47 13 5 1171 227 11 3 163 79 140+ 3 1231 3 None 5 7 3 1201 7 3 13 >10000 3 5 3 7 17 7 13 7 160+ 3 3 7 3 5 137 3 3 None 17 181 5 3 3251 5 3 5 347 19 7 180+ 17 167 223 >10000 >10000 7 37 3 3 13 17 3 5 3 11 None 31 5 577 >10000 200+ 271 37 3 5 19 3 13 5 3 >10000 41 11 137 191 3 None 281 3 13 7 220+ 7 5 239 11 None 127 5 461 11 5333 3 953 113 61 7 3 7 7 5 109 240+ 17 19 None 3331 3 3 17 41 5 127 7 541 19 5 5 None 23 11 2011 5 260+ 31 197 5 7 5 3 13 11 >10000 241 41 3 37 5 5 31 5 3 3 7 280+ >10000 7 29 2473 5 13 3 3 None 3 13 5 3 7 17 41 17 53 113 7

There are only probable primes for that b = 18, 51, 91, 96, 174, 230, 244, 259, and 284.

No known repunit primes or PRPs for that b = 152, 184, 185, 200, 210, 269, and 281.

Because of the algebra factorization, there are no repunit primes for that b = 4, 9, 16, 25, 32, 36, 49, 64, 81, 100, 121, 125, 144, 169, 196, 216, 225, 243, 256, and 289.

For negative bases (up to -300), see Wagstaff prime.

### The smallest natural number base b that $R_p$ is prime for prime p

The list is about the first 100 primes. (sequence A066180 in OEIS)

 p 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 min b 2 2 2 2 5 2 2 2 10 6 2 61 14 15 5 24 19 2 46 3 p 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 min b 11 22 41 2 12 22 3 2 12 86 2 7 13 11 5 29 56 30 44 60 p 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 min b 304 5 74 118 33 156 46 183 72 606 602 223 115 37 52 104 41 6 338 217 p 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 min b 13 136 220 162 35 10 218 19 26 39 12 22 67 120 195 48 54 463 38 41 p 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 min b 17 808 404 46 76 793 38 28 215 37 236 59 15 514 260 498 6 2 95 3

### List of repunit primes base b

 b n which $R_n(b)$ is prime (numbers greater than 232768 are only PRP, these n's are checked up to 100000) OEIS sequence −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 (Algebra) −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 A057186 −19 17, 37, 157, 163, 631, 7351, 26183, 30713, 41201, 77951 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 A057175 −8 2 (Algebra) −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 (Aurifeuillean factorization) −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 A007658 −2 3, 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 (1 is not prime) A000978 −1 None (1 is not prime) 0 None (1 is not prime) 1 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, ... (All primes) A000040 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 (Algebra) 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 (Algebra) 9 None (Algebra) 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 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 A006032 15 3, 43, 73, 487, 2579, 8741, 37441, 89009 A006033 16 2 (Algebra) 17 3, 5, 7, 11, 47, 71, 419, 4799, 35149, 54919, 74509 A006034 18 2, 25667, 28807, 142031, 157051, 180181 A133857 19 19, 31, 47, 59, 61, 107, 337, 1061, 9511, 22051, 209359 A006035 20 3, 11, 17, 1487, 31013, 48859, 61403 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 (Algebra) 26 7, 43, 347, 12421, 12473, 26717 A127999 27 3 (Algebra) 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

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

If b is a power (can be written as mn, with m, n integers, n > 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 $R_p$, $R_p$, 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. if n is not a prime power, then no base b repunit exists, for example, b = 64 (with n = 6), and b = -1, 0, or 1 (with n can be any natural number). Another special situation is b = -4, which has the aurifeuillean factorization $4^n+1 = 4^{2k-1}+1 = (2^{2k-1}-2^k+1) (2^{2k-1}+2^k+1)$.There is also a conjecture that when b is neither a power nor -4, then there are infinity many base b repunit primes.

## 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 recurring 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 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

The Demlo numbers[9] 1, 121, 12321, 1234321, … 12345678987654321, 1234567900987654321, 123456790120987654321, … 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.