A note on Dujella's unicity conjecture

Abstract

Using properties of binary quadratic Diophantine equations, we prove that if \(r=p^{m} q^{n}\), where \(p, q\) are distinct odd primes and \(m, n\) are positive integers, then the equation \(x^{2}-\left(r^{2}+1\right) y^{2}=r^{2}\) has at most one positive integer solution \((x, y)\) with \(y \lt r-1\).