Does there exist any integer $ n> 1$ for which $6^{2n}-25$ is prime?

Question

I got this question on a test and I am really curious hoe you would approach it. I tried to prove stuff using the congruence laws but I didn't manage to prove anything.

Answer 1

We have $$N=6^{2n}-25=(6^n-5)(6^n+5)$$ none of these factors equal $1$(unless for $n=1$), Therefore $N$ is always composite.

Answer 2

Hint: You are looking at a difference of squares.