1. ## Congruence

Find all integer solutions to
$-2^0-2^1-2^2-...-2^{q-1} \equiv 0 \bmod q^{p-1}$
Where
$2p \geq q$
$p$ is prime.

I know that the only solution is 1,1 (Using brute force through my graphing calculator)
This is actually part of another question in which I tried to solve and got stuck here. (I'm pretty sure what I've done before this is correct)
So if $p$ being prime is not used, it's alright.

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Then you are asking us to solve the congruence, given arbitrary q, find a prime $p \geq q/2$ such that: $q^{p-1}|2^q-1$