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A bound on perfect squares as values of a polynomial in m

Source: 4th Memorial Mathematical Competition "Aleksandar Blazhevski - Cane"- Junior D1 P2/ Senior D1 P1

December 20, 2022
number theorypolynomialfloor functionPerfect Square

Problem Statement

Let a,b,c,da, b, c, d be integers. Prove that for any positive integer nn, there are at least ⌊n4⌋\left \lfloor{\frac{n}{4}}\right \rfloor positive integers m≤nm \leq n such that m5+dm4+cm3+bm2+2023m+am^5 + dm^4 + cm^3 + bm^2 + 2023m + a is not a perfect square.
Proposed by Ilir Snopce
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