# Prime Pickle

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

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