Repunit

A repunit is a positive integer that consists entirely of the digit 1 in a given base. In base 10, the decimal repunits are numbers such as 1, 11, 111, and so on. More generally, a repunit in base b with n digits is the number consisting of n consecutive 1s in base b.

Notation and formula: R_n(b) denotes this number. It can be written as R_n(b) = 1 + b + b^2 + ... + b^{n-1} = (b^n - 1)/(b - 1). In base 2, R_n(2) = 2^n - 1, which are the Mersenne numbers.

Properties: If d divides n, then R_d(b) divides R_n(b). This follows from the factorization of b^n - 1. Also gcd(R_n(b), b) = 1 for any n > 0, since R_n(b) ≡ 1 mod b. The decimal repunits end in 1 and are thus coprime to 10.

Primality and study: A repunit prime is a repunit that is prime. In base 10, a necessary condition for R_n(10) to be prime is that n is prime, but this is not sufficient; only a few such primes are known and it is unknown whether infinitely many exist. The search for large repunit primes is a matter of computational number theory.

See also: Mersenne numbers.