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Fibonacci divisibilities

Source: 2014 BAMO-12 #4

February 22, 2016
FibonacciSequenceDivisibilitynumber theory

Problem Statement

Let F1,F2,F3⋯F_1, F_2, F_3 \cdots be the Fibonacci sequence, the sequence of positive integers satisfying F1=F2=1F_1 =F_2=1 and Fn+2=Fn+1+FnF_{n+2} = F_{n+1} + F_n for all n≥1n \ge 1.
Does there exist an n≥1n \ge 1 such that FnF_n is divisible by 20142014? Prove your answer.
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