If n is even, we claim that 14^{n} + 11 is divisible by 3.
Let n = 2k, and working mod 3 we have:
14^{n} + 11 
= 
14^{2k} + 11 

= 
2^{2k} + 2 (mod 3) 

= 
4^{k} + 2 (mod 3) 

= 
1^{k} + 2 (mod 3) 

= 
1 + 2 (mod 3) 

= 
0 (mod 3) 
If n is odd, we claim that 14^{n} + 11 is divisible by 5.
Let n = 2k + 1, and working mod 5 we have:
14^{n} + 11 
= 
14^{2k+1} + 11 

= 
4^{2k+1} + 1 (mod 5) 

= 
4.16^{k} + 1 (mod 5) 

= 
4.1^{k} + 1 (mod 5) 

= 
4 + 1 (mod 5) 

= 
0 (mod 5) 
Hence, for any n >= 0, 14^{n} + 11 is divisible by 3 or 5,
and is not prime.