A repunit is a number whose decimal representation contains only the digit 1. The \(n\)-digit repunit is \(R_n = (10^n - 1)/9\): 1, 11, 111, 1111, 11111.
Most repunits are composite, but some are prime — these are the repunit primes. The known indices for prime repunits are \(n = 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, \ldots\), all rare.
Repunits factor in special ways: \(R_{ab}\) is always divisible by \(R_a\) and \(R_b\), so for a repunit to be prime its index must be prime. (The converse fails: \(R_3 = 111 = 3 \cdot 37\) is composite even though 3 is prime.)