MathDB
Tricky Sequence With NT Properties

Source: 2019 China Second Round P3

September 8, 2019
China second roundalgebranumber theory

Problem Statement

Let mm be an integer where ∣m∣≥2|m|\ge 2. Let a1,a2,⋯a_1,a_2,\cdots be a sequence of integers such that a1,a2a_1,a_2 are not both zero, and for any positive integer nn, an+2=an+1−mana_{n+2}=a_{n+1}-ma_n.
Prove that if positive integers r>s≥2r>s\ge 2 satisfy ar=as=a1a_r=a_s=a_1, then r−s≥∣m∣r-s\ge |m|.
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