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determine k(2),k(4),k(5),k(10)

Source: 2022 Viet Nam math olympiad for high school students D2 P4

March 22, 2023
algebra

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. a) Prove that: For all m1,m2∈Z+m_1,m_2\in \mathbb{Z^+}, we have:k([m1,m2])=[k(m1),k(m2)].k([m_1,m_2])=[k(m_1),k(m_2)].(Here [a,b][a,b] is the least common multiple of a,b.a,b.)
b) Determine k(2),k(4),k(5),k(10).k(2),k(4),k(5),k(10).
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