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Fibonacci sequence, congruent mod p

Source: 2022 Viet Nam math olympiad for high school students D2 P5

March 22, 2023
algebranumber theory

Problem Statement

Given Fibonacci sequence (Fn),(F_n), and a positive integer mm, denote k(m)k(m) by the smallest positive integer satisfying Fn+k(m)≡Fn( mod m),F_{n+k(m)}\equiv F_n(\bmod m), for all natural numbers nn, pp is an odd prime such that p≡±1( mod 5)p \equiv \pm 1(\bmod 5). Prove that: a) 5p−12≡1( mod p).{5^{\frac{{p - 1}}{2}}} \equiv 1(\bmod p). b) Fp−1≡0( mod p).{F_{p - 1}} \equiv 0(\bmod p). c) k(p)∣p−1.k(p)|p-1.
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