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Fibonacci sequence, congruent, k(2^s)

Source: 2022 Viet Nam math olympiad for high school students D2 P7

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, ss is a positive integer. Prove that: a) F3.2s−1≡0( mod 2s){F_{{{3.2}^{s - 1}}}} \equiv 0(\bmod {2^s}) and F3.2s−1+1≡1( mod 2s).{F_{{{3.2}^{s - 1}} + 1}} \equiv 1(\bmod {2^s}). b) k(2s)=3.2s−1.k({2^s}) = {3.2^{s - 1}}.
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