Prime Pickle

Find the number of positive integers m100m\leq 100 such that for at least 500500 values of positive integers n,n, 1n1000,1\leq n \leq 1000, there exist a prime pp and integers aa and bb such that {anamb(modp)bna(modp)(a2a)(b2b)≢0(modp)\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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