MathDB

6

Part of IMSC 2024

Problems(1)

Expression is a perfect square implies the polynomial is constant

Source: IMSC Day 2 Problem 3

6/29/2024
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
algebrapolynomialnumber theorynumber theory proposedalgebra proposedLifting the ExponentDivisibility