Glasnik Matematicki, Vol. 46, No. 2 (2011), 325-331.

ON A VARIANT OF A DIOPHANTINE EQUATION OF CASSELS

Alain Togbé and Pingzhi Yuan

Mathematics Department, Purdue University North Central, 1401 S, U.S. 421, Westville IN 46391, USA
e-mail: atogbe@pnc.edu

School of Mathematics, South China Normal University, Guangzhou 510631, P. R. China
e-mail: mcsypz@mail.sysu.edu.cn


Abstract.   Recently, Yuan and Li considered a variant y2=px(Ax2-2) of Cassels' equation y2=3x(x2+2). They proved that the equation has at most five solutions in positive integers (x, y). In this note, we improve Yuan-Li's result by showing that for any prime p and any odd positive integer A, the Diophantine equation y2=px(Ax2-2) has at most three solutions in positive integers (x, y).

2000 Mathematics Subject Classification.   11D25, 11B39.

Key words and phrases.   Diophantine equations.


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DOI: 10.3336/gm.46.2.04


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