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Let n be a positive integer, let a_0 = 1, and suppose that for every positive integer i the number a_i is the residue modulo n of (a_i-1)^2 - 1. Suppose that p is a prime divisor of n.
(a) Show that there exist integers i and j such that 0 <= i < j < p such that a_i = a_j (mod p).
(b) Show that if a_i = a_j (mod p) then a_i+k = a_j+k (mod p) for all positive integers k.
(c) Let i and j be as in (a), and suppose that k is a multiple of j - i with i <= k < j. Show that a_2k = a_k (mod p).
I cannot do (a), but can do part (b) and (c) assuming (a)...
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