$2^{8420} - 9$ is prime or not

Question

How to prove $2^{8420} - 9$ is or isn't a prime number?

I tried modding it by 10 to get the last digit, but that's a 7 which doesn't help.

We've only been covering successive squaring in this chapter (and modular arithmetic).

Answer 1

Hint: Use the factorization $a^2-b^2 = (a-b)(a+b)$ for a useful choice of values for $a$ and $b$.

When you do this, it should be fairly obvious that both factors are bigger than $1$.

Answer 2

The number is a multiple of $13$ because $2^{12} \equiv 1 \bmod 13$, $8240 \equiv 8 \bmod 12$, and $2^8 \equiv 9 \bmod 13$.