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Expression is a perfect square implies the polynomial is constant

Source: IMSC Day 2 Problem 3

June 29, 2024
algebrapolynomialnumber theorynumber theory proposedalgebra proposedLifting the ExponentDivisibility

Problem Statement

Let a≡1(mod4)a\equiv 1\pmod{4} be a positive integer. Show that any polynomial Q∈Z[X]Q\in\mathbb{Z}[X] with all positive coefficients such that Q(n+1)((a+1)Q(n)−aQ(n))Q(n+1)((a+1)^{Q(n)}-a^{Q(n)}) is a perfect square for any n∈N∗n\in\mathbb{N}^{\ast} must be a constant polynomial.
Proposed by Vlad Matei, Romania
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