Use Euler's Criterion to deduce that $5^{2^{2^{n}-1}}\equiv -1 \space \text{mod} \space p $

Question

Let $p$ be a prime and suppose $$p=2^{2^{n}}+1$$ for some $n \ge 2$.

Use Euler's Criterion to deduce that $$5^{2^{2^{n}-1}}\equiv -1 \space \text{mod} \space p $$

I am not sure how Euler's criterion would help in this situation, even though I have been told to use it.

Answer 1

Applying Euler's criterion reduces the problem to showing that $5$ is not a quadratic residue mod $p$

Then, using Gauss' reciprocity law reduces to showing that $p$ is not a quadratic residue modulo $5$

This is easy to show since,

$$p=2^{2^n}+1\equiv 2^{2^n\bmod 4}+1\equiv 2^0+1\equiv 2\pmod 5$$

and $2$ is not a quadratic residue mod $5$ (only $0,1,4$ are).