Quadratic residue modulo odd power of $2$

Question

If $x$ and $n$ both are odd positive integers, such that, $$x^2 \equiv -1\mod2^n$$ what can we say about $x$ and $n$ ?

Answer 1

Since odd square $ \equiv 1 \pmod 8$ we infer that if $n \geq 3$, $x^2+1 \equiv 2 \neq -1 \pmod 8$. Thus $n=1$.

Answer 2

Hint: Note that if $x$ is odd, then $x^2+1\equiv 2\pmod{4}$.