Problem

Source: FKMO 2021 Problem 2

Tags: Diophantine equation, number theory, FKMO



Positive integer k(8) is given. Prove that if there exists a pair of positive integers (x,y) that satisfies the conditions below, then there exists infinitely many pairs (x,y). (1) xy23,yx22 (2) gcd(3x+2(y23)x,2y+3(x22)y)=k