Article

On the simultaneous Pell equations x² − (4m² − 1)y² = y² − pz² = 1

Published in: Quaestiones Mathematicae
Volume 40 , issue 5 , 2017 , pages: 697–703

DOI: 10.2989/16073606.2017.1310145

Author(s): Tingting Wang College of Science, P.R.China , Yingzhao Jiang College of Science, P.R.China

Keywords: 11D09 , 11D09

Abstract

Let m be a positive integer, and let p be an odd prime. By using certain properties of Pell and quartic diophantine equations with some elementary number theory methods, we prove that the system of equations x² − (4m² − 1)y² = 1 and y² − pz² = 1 has positive integer solutions (x, y, z) if and only if p ≡ 7(mod 8) and , where (f, g) is a positive integer solution of the equation f ² −pg² = 2. Further, if the above condition is satisfied, then the system of equations has only the positive integer solution .

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