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0473 Fibonacci 4th edition Round 7 p3

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May 7, 2021
algebra4th edition

Problem Statement

Let {fn}n≥0\{f_n\}_{n \ge 0} be the Fibonacci sequence, given by f0=f1=1f_0 = f_1 = 1, and for all positive integers nn the recurrence fn+1=fn+fn−1f_{n+1} = f_n + f_{n-1}. Let an=fn+1fna_n = f_{n+1}f_n for any non-negative integer nn, and let Pn(X)=Xn+an−1Xn−1+...+a1X+a0.P_n(X) = X^n + a_{n-1}X^{n-1} + ... + a_1X + a_0. Prove that for all positive integers n≥3n \ge 3 the polynomial Pn(X)P_n(X) is irreducible in Z[X]Z[X].
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