A repunit is a number consisting solely of ones (such as 11 or 11111).
Let us call p(n) a 10-base integer represented by a string of n ones, e.g. p(1)=1, p(5)=11111 etc.
Most of the repunit numbers are composite.
2, 19,23,317 are the first four indices of prime repunits.

Prove: For a prime repunit p(n) to be prime, n has to be prime.

If n is composite, then n 1's is composite. Say n=xy. Then, n 1's is divisible by x 1's and y 1's. Therefore, if n 1's is prime, then n is prime.

Posted by Math Man on 2011-01-10 17:40:22