Let $a\in \Bbb Z$, $b\in \Bbb Z$ such that $p \nmid b$, and $p$ a prime where $p \gt 2$.
If for all $x \in \Bbb Z$ such that $p \nmid x$ and $\operatorname{ord}_p(x) \ne p-1$, $p$ satisfies $\operatorname{ord}_p(a+bx) = p-1$, prove that $p$ is in the form:
$p = 2^{2^n} + 1$ for some $n$ non-negative.