Denote $a = 11114$, $p = 44449$, $q=21433$ and note that $p$ and $q$ are primes ($a$ isn't prime).
I wish to find a natural number $n$ such that : $a^n \equiv q\mbox{ mod }p$.
I tried to find such an $n$ but I couldn't do it... I think it has something to do with Fermat's little theorem. I also tried writing $a^n-q=kp$ (for $k \in \mathbb{Z}$) and then looked at the equation mod $a$ and mod $q$ to try and gain more information but that didn't lead me to an answer as well.