Problem

Source: 2023 Indonesia TST Stage 1: Test 1 - Number Theory

Tags: number theory, GCD, Integer sequence, infinite sequence, coprime



Given an integer $a>1$. Prove that there exists a sequence of positive integers \[ n_1, n_2, n_3, \ldots \]Such that \[ \gcd(a^{n_i+1} + a^{n_i} - 1, \ a^{n_j + 1} + a^{n_j} - 1) =1 \]For every $i \neq j$.