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F_n + 2 = F_{n+1} + 1 = F_{n+2} mod m, Fibonacci

Source: 2013 Saudi Arabia GMO TST II p4

July 26, 2020
fibonacci numberremaindernumber theory

Problem Statement

Let F0=0,F1=1F_0 = 0, F_1 = 1 and Fn+1=Fn+Fn−1F_{n+1} = F_n + F_{n-1}, for all positive integer nn, be the Fibonacci sequence. Prove that for any positive integer mm there exist infinitely many positive integers nn such that Fn+2≡Fn+1+1≡Fn+2F_n + 2 \equiv F_{n+1} + 1 \equiv F_{n+2} mod mm .
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