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

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

Thus, the number R_n consists of n copies of the digit 1. The sequence of repunits starts 1, 11, 111, 1111,... (sequence A002275 in OEIS).

Repunit primes

Historically, the definition of repunits was motivated by recreational mathematicians looking for prime factors of such numbers. Wikipedia contains a list of repunit factorizations.

It is easy to show that if n is divisible by a, then R_n is divisible by R_a. For example, 9 is divisible by 3, and indeed R₉ is divisible by R₃—in fact, 111111111 = 111 · 1001001. 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.

Repunit primes turn out to be rare. R_n is prime for n = 2, 19, 23, 317, 1031,... (sequence A004023 in OEIS). R₄₉₀₈₁ and R₈₆₄₅₃ are probably prime. It has been conjectured that there are infinitely many repunit primes.

Generalizations

Professional mathematicians used to consider repunits an arbitrary concept, arguing that it depends on the use of decimal numerals. But the arbitrariness can be removed by generalizing the idea to base-b repunits:

$R_{n}^{(b)}={b^{n}-1 \over b-1}\qquad {\mbox{for }}n\geq 1.$

In fact, the base-2 repunits are the well-respected Mersenne numbers M_n = 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.

The prime repunits are a subset of the permutable primes, i.e., primes that remain prime after any permutation of their digits.

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.

See also

• Repdigit
• Table of Factors

External links

Web sites

• Repunit article on 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.

Books

• A. Beiler, Recreations in the theory of numbers. ISBN 0486210960. Chapter 11, of course.
• Paulo Ribenboim, The New Book Of Prime Number Records. ISBN 0387944575.