A note on the Diophantine equation y(2) = px (Ax(2) +2)

Recently, Li Min Chen (Acta Math Sinica Chin Ser 53:83-86, 2010) considered a variant y(2) = px (x(2) + 2) of Cassel's 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 Cassel's 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).