Glasnik Matematicki, Vol. 58, No. 1 (2023), 59-65. \( \)

A NOTE ON DUJELLA'S UNICITY CONJECTURE

Maohua Le and Anitha Srinivasan

Institute of Mathematics, Lingnan Normal College, Zhangjiang, Guangdong, 524048, China
e-mail:lemaohua2008@163.com

Departamento de métodos cuantitativos, Universidad Pontificia de Comillas (ICADE), C/ Alberto Aguilera, 23 - 28015, Madrid, Spain
e-mail:asrinivasan@icade.comillas.edu

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 \(y \lt r-1\).

2020 Mathematics Subject Classification.   11D09, 11R29, 11E16

Key words and phrases.   Binary quadratic forms, quadratic diophantine equation, Dujella's conjecture

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https://doi.org/10.3336/gm.58.1.04

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