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Number theory and Fibonacci

Source: 2016 Taiwan TST Round 2

July 18, 2016
number theoryFibonacci sequence

Problem Statement

Let <Fn>\left< F_n\right> be the Fibonacci sequence, that is, F0=0F_0=0, F1=1F_1=1, and Fn+2=Fn+1+FnF_{n+2}=F_{n+1}+F_{n} holds for all nonnegative integers nn. Find all pairs (a,b)(a,b) of positive integers with a<ba < b such that Fn−2nanF_n-2na^n is divisible by bb for all positive integers nn.
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