repunit

Given base $b$, a number of the form $\frac{b^n-1}{b-1}$ for $n>0$ is written using using only the digit 1 in that base and is therefore a repunit. The term, short for ”repeated unit,” is credited to Beiler’s book Recreations in the theory of numbers, in chapter 11.

Regardless of base, a prime number is a prime number, but if in a given base it is a repunit, then it is called a repunit prime in that base. In binary, the Mersenne numbers are repunits, therefore the Mersenne primes are repunit primes in binary. Repunit primes in base 10 appear to be fewer, with only seven known as of 2006, for $n$ taking on the values 2, 19, 23, 317, 1031, 49081, 86453 (see Sloane’s OEIS, A004023, for updates).

In a trivial way, repunit primes are also palindromic primes and permutable primes.

Title	repunit
Canonical name	Repunit
Date of creation	2013-03-22 16:13:26
Last modified on	2013-03-22 16:13:26
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A63
Defines	repunit prime