Let u(α,b) be a Lucas sequence satisfying the second-order recursion relation u_(n+2) = αu_(n+1) + bu_n, where b = ±1,α is an integer, and u_0 = 0 and u_1 = 1. Let m be a positive integer, and let π(m) denote the period of u(α, b) modulo m, and p(m) denote the restricted period of u(a,b) modulo m. It is shown that iterates of π(m) and p(m) end in either a fixed point or a cycle of length two, and all these possible fixed points and two-cycles are explicitly determined.
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