11₂=3 and 111₃=13 are prime. However, 1111₄=85=5*17 and 11111₅=781=11*71 are not prime. What is the least number n>3 such that the repunit consisting of n 1's in base n is prime?

The general formula is x=(n^n-1)/(n-1), which gives a base 10 value that can quickly be checked for primality. Obviously, only prime n need be checked. Using this method,

{{n == 2, x == 3}

 {n == 3, x == 13}

 {n == 5, x == 781}

 {n == 7, x == 137257}

 {n == 11, x == 28531167061}

 {n == 13, x == 25239592216021}

 {n == 17, x == 51702516367896047761}

 {n == 19, x == 109912203092239643840221}...}

109912203092239643840221 is prime, and in base 19 is:

1111111111111111111.

Edited on July 24, 2013, 11:45 am

Posted by broll on 2013-07-24 11:29:46