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 \[\begin{cases} a^n\equiv a^m\equiv b \pmod p\\ b^n\equiv a \pmod p\\ (a^2-a)(b^2-b) \not \equiv 0 \pmod p \end{cases}\]