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Find the number of positive integers m≤100m\leq 100m≤100 such that for at least 500500500 values of positive integers n,n,n, 1≤n≤1000,1\leq n \leq 1000,1≤n≤1000, there exist a prime ppp and integers aaa and bbb such that {an≡am≡b(modp)bn≡a(modp)(a2−a)(b2−b)≢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}⎩⎪⎨⎪⎧an≡am≡b(modp)bn≡a(modp)(a2−a)(b2−b)≡0(modp)
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