A note on the Diophantine equation $$y^2=px(A{x^{2}}+2)$$

Abstract

Recently, Li Min Chen (Acta Math Sinica Chin Ser 53:83–86, 2010) considered a variant \(y^2=px(x^2+2)\) of Cassels’ equation \(y^2=3x(x^2+2)\). He proved that the equation has at most two solutions in positive integers \((x, y)\). Therefore, he improved a result obtained Luca and Walsh (Glasgow Math J 47:303–307, 2005). In this note, we consider another variant of Cassels’ equation and we show that for any prime \(p\) and any odd positive integer \(A\), the Diophantine equation \(y^2=px(Ax^2+2)\) has at most seven solutions in positive integers \((x, y)\).