Products of consecutive integers and the Markoff equation
\textit{S. Katayama} [Proc. Japan Acad., Ser. A 66, 305-306 (1990; Zbl 0755.11011)] described all integral solutions to the diophantine equation \(X(X+ 1) Y(Y+ 1)= Z(Z+ 1)\). The author clarifies the description by using a bijection studied by Mordell with \(x^2+ y^2+ z^2= 2xyz+ 5\). Also, it is shown that the number of positive integer solutions with \(Z\leq H\) to Katayama's equation is of order \(\sqrt H\), and, in general, the number of solutions to Mordell's equation \(x^2+ y^2+ z^2= axyz+ b\) with height less than \(H\) is counted.
number of solutions to Mordell's equation
scientific article; zbMATH DE number 856830