Problem

Source: ELMO Shortlist 2023 N5

Tags: Elmo, number theory



An ordered pair \((k,n)\) of positive integers is good if there exists an ordered quadruple \((a,b,c,d)\) of positive integers such that \(a^3+b^k=c^3+d^k\) and \(abcd=n\). Prove that there exist infinitely many positive integers \(n\) such that \((2022,n)\) is not good but \((2023,n)\) is good. Proposed by Luke Robitaille