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Problem 971

AMC 12 late, AIME early

Number theory Difficulty 4.8 Prove it Round 3 · Mongolia

If $m$ and $n$ are positive integers, can the number $m^4 + 2mn + n^2 - 2021$ be the product of three or more consecutive integers?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Answer: No.
Suppose that $N = m^4 + 2mn + n^2 - 2021$ is the product of three or more consecutive integers. Then $N$ and $m^4 - m^2 = m[(m-1)m(m+1)]$ are divisible by $3$ . Thus $(m+n)^2 \equiv N - (m^4 - m^2) + 2021 \equiv 2 \pmod{3}$ , which is a contradiction, since $2$ is not a quadratic residue modulo $3$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.