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Israel 2016 Q5 - Fibonacci valued polynomial

Source: Israel National Olympiad 2016 Q5

August 7, 2019
algebranumber theoryFibonacciFibonacci sequencePolynomialsdegreepolynomial

Problem Statement

The Fibonacci sequence FnF_n is defined by F1=F2=1F_1=F_2=1 and the recurrence relation Fn=Fn−1+Fn−2F_n=F_{n-1}+F_{n-2} for all integers n≥3n\geq3.
Let m,n≥1m,n\geq1 be integers. Find the minimal degree dd for which there exists a polynomial f(x)=adxd+ad−1xd−1+⋯+a1x+a0f(x)=a_dx^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0, which satisfies f(k)=Fm+kf(k)=F_{m+k} for all k=0,1,...,nk=0,1,...,n.
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