Repunits and prime numbers

A repunit (“repeated unit”) is a number that only contains the digit 1 in some number base. For example, 31 is a repunit in base 5 and base 2, because 31 is written as 111 (25+5+1) in base 5, and as 11111 (16+8+4+2+1) in base 2. Another number that is a repunit in two different bases is 8191, which is 111 in base 90 and 1111111111111 in base 2.