Let $p>5$ be a prime, $n\triangleq\frac{4^p+1}{5}\;\quad\text{and}\;\quad b\triangleq\frac{n-1}{4}\quad.$
It can be shown that both are integers, and also that $n$ is composite, $b$ is odd and $p$ divides $b$.
I'd like to prove that $4^b\equiv-1\pmod n\;\quad.$
I tried rephrasing the congruence in terms of the other variables to see if I could find a way out by applying Fermat's — or even Euler's — Theorem, but I'm not getting anywhere.