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# Repunit

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by A.H. Beiler. A repunit prime is a repunit that is also a prime number.

## Definition

The repunits are defined mathematically as
$R_n=\left\{10^n-1over9\right\}qquadmbox\left\{for \right\}nge1.$
Thus, the number Rn consists of n copies of the digit 1. The sequence of repunits starts 111111, 1111,... (sequence A002275 in OEIS).

## 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.

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 and it is curious that, though Edouard 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 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.

## 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 is divisible by Ra:


R_n=frac{1}{9}prod_{d|n}Phi_d(10)

where $Phi_d$ is the $d^mathrm\left\{th\right\}$ cyclotomic polynomial and d ranges over the divisors of n. For p prime, $Phi_p\left(x\right)=sum_\left\{i=0\right\}^\left\{p-1\right\}x^i$, which has the expected form of a repunit when x is substituted for with 10.

For example, 9 is divisible by 3, and indeed R9 is divisible by R3—in fact, 111111111 = 111 · 1001001. The corresponding cyclotomic polynomals $Phi_3\left(x\right)$ and $Phi_9\left(x\right)$ 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.

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. He later announced there are no others from R86453 to R200000. On July 15 2007 Maksym Voznyy announced R270343 to be probably prime , along with his intent to search to 400000.

It has been conjectured that there are infinitely many repunit primes 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.

## Generalizations

Professional mathematicians used to consider repunits an arbitrary concept, since they depend on the use of decimal numerals. But the arbitrariness can be removed by generalizing the idea to base-b repunits:
$R_n^\left\{\left(b\right)\right\}=\left\{b^n-1over b-1\right\}qquadmbox\left\{for \right\}nge1.$

In fact, the base-2 repunits are the well-respected Mersenne numbers Mn = 2n − 1. 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.

Example 1) 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).

Example 2) the only base-4 repunit prime is 5 ($11_4$), because $4^n-1=left\left(2^n+1right\right)left\left(2^n-1right\right)$, and 3 divides one of these, leaving the other as a factor of the repunit.

It is easy to prove that given n, such that n is not exactly divisible by 2 or p, there exists a repunit in base 2p that is a multiple of n.

## References

Web sites

Books

• S. Yates, Repunits and repetends. ISBN 0-9608652-0-9.
• A. Beiler, Recreations in the theory of numbers. ISBN 0-486-21096-0. Chapter 11, of course.
• Paulo Ribenboim, The New Book Of Prime Number Records. ISBN 0-387-94457-5.

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