Number Theory Problem Using Group Theory

Question

So I have a problem "Let $a,b\in \mathbb{N}$ satisfy $\text{gcd}(a,b)=1$ and $a^{2^n}+b^{2^n} \equiv 0 \pmod p$, in which $p$ is prime. Prove $2^{n+1}|(p-1)$". I approach this problem using group theory, we can prove this if $2^{n+1}$ is the order of $a$, but it doesn't seem right? Help me

Answer 1

Since $(a,b) =1$, at least one of them is coprime to $p$, so without loss of generality, $(b,p) = 1$. Hence, we can rewrite the equation as $$(a/b)^{2^n}\equiv -1\pmod p,$$

so $$(a/b)^{2^{n+1}}\equiv 1\pmod p.$$

What can you deduce about the order of $\frac ab$?