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Abstract
Recently, Katayama described all integral solutions to the diophantine equationX(X + 1)Y(Y + 1) = Z(Z + 1). In this paper, we clarify his description by noting a bijection withx 2 + y 2 + z 2 = 2xyz + 5, which has been studied by Mordell. We also show the number of positive integer solutions withZ ≤H to Katayama's equation is of order\(\sqrt H \), and in general, count the number of solutions to Mordell's equationx 2 + y 2 + z 2 = axyz + b with height less thanH.
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