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

(In reply to Answer by K Sengupta)

Let us suppose that n is even. Then, n = 2s, for some positive integer s.

Then, 11*14^n + 1 = 11*196^s + 1

Now, 196 (mod 3) = 1
-> 196^s (mod 3) = 1, so that:
196^s = 3*x + 1, for some integer x.
-> 11*196^s + 1 = 33x + 12 = 3(11x + 4), so that 11*196^s + 1 is always divisible by 3, for every value of s.

Thus, 11 x 14^n + 1 is always composite whenever n is even.
 
Let us suppose that n is odd. If so, the last digit of 14^n will always be 4, so that the unit digit of 11* 14^n + 1 is 5, so that 11* 14^n + 1 is divisible by 5, and accordingly the given expression  cannot be a prime.

Combining the above two cases, it follows that: 11*14^n + 1 will never be a prime number. 

Posted by K Sengupta on 2008-11-21 00:41:34