Let $x$ be an odd prime, and let $a$ and $b$ be positive integers, with $a \gt 1$.
Suppose $a^{2^{b}} \equiv −1 \pmod{x}$. Then $x \equiv 1 \pmod{2^{b+1}}$.
I have to solve using number theory concepts - it's homework for number theory.
I start by noting that: $a^{2^{b+1}} ≡ 1 \pmod{x}$
But I don't know where to go next.. I don't think I'm allowed to use concepts like the order of an element in a group (since we haven't even talked about groups in the class yet). Would greatly appreciate some direction, hints, anything!