Repunit

A repunit is a number like 11, 111, or 1111. It only has the digit 1 in it. It is a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book "Recreations in the Theory of Numbers".
A repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes.
Definition.
The base-"b" repunits can be written as this where "b" is the base and "n" is the number that you are checking in whether or not it is a repunit:
This means that the number "R""n"("b") is made of of of "n" copies of the digit 1 in base-"b" representation. The first two repunits base-"b" for "n" = 1 and "n" = 2 are 
The first of repunits in base-10 are with
Base-2 repunits are also Mersenne numbers "M""n" = 2"n" − 1. They start with
Factorization of decimal repunits.
Prime factors that are are "new factors" that haven't been mentioned before. Basically, the prime factor divides "R""n" but does not divide "R""k" for all "k" &lt; "n." 
The smallest prime factors of "R""n" for "n" &gt; 1 are