# 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.

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 relative 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 10i = 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 = 3 · 3 · 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 = 3 · 3 · 7 · 11 · 13 · 19 · 37 · 52579 · 333667 R19 = 1111111111111111111 R20 = 11 · 41 · 101 · 271 · 3541 · 9091 · 27961

## 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.[1] He later announced there are no others from R86453 to R200000.[2] On July 15, 2007 Maksym Voznyy announced R270343 to be probably prime,[3] 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[4] 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 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 (quinary) 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 (duodecimal) 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 (vigesimal) 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

## 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.[5]

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[6] 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[7] 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[8] 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.