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Fibonacci sequence, k(p), congruent

Source: 2022 Viet Nam math olympiad for high school students D2 P6

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) Fp+1≡0( mod p).{F_{p + 1}} \equiv 0(\bmod p). b) k(p)∣2p+2.k(p)|2p+2. c) k(p)k(p) is divisible by 4.4.
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