Find the smallest integer n that makes 11 x 14^n + 1, a prime number, or, prove that it doesn't exist.

There doesn't exist such a number.

First, consider the case where n is odd. In this case, 14^n mod 10 will always be 4. 14^n*11 mod 10 will then also be 4, and 11*14^n+1 mod 10 = 5, so this total number will be divisible by 5, and can't be equal to 5.

Now consider the case where n is even, n = 2*m.
Notice that
11 mod 3 = 2
14 mod 3 = 2
14^2 mod 3 = 1
(14^2)^m mod 3 = 1
11*14^(2*m) mod 3 = 2
11*14^(2*m) + 1 mod 3 = 0
so the total number will always be divisible by 3, and not equal to 3.

Posted by Robby Goetschalckx on 2005-06-08 14:00:19