Problem

Source: Vietnam TST 2023 P4

Tags: number theory



Given are two coprime positive integers a,b with b odd and a>2. The sequence (xn) is defined by x0=2,x1=a and xn+2=axn+1+bxn for n1. Prove that: a) If a is even then there do not exist positive integers m,n,p such that xmxnxp is a positive integer. b) If a is odd then there do not exist positive integers m,n,p such that mnp is even and xmxnxp is a perfect square.