Prime Pickle

Number Theory Level 5

Find the number of positive integers $m\leq 100$ such that for at least $500$ values of positive integers $n,$ $1\leq n \leq 1000,$ there exist a prime $p$ and integers $a$ and $b$ such that $\begin{cases} a^n\equiv a^m\equiv b \pmod p\\ b^n\equiv a \pmod p\\ (a^2-a)(b^2-b) \not \equiv 0 \pmod p \end{cases}$