A system of cubic diophantine equations

Abstract

In this paper we solve x 3 + y + 1 − xyz = 0 completely and study a pair of simultaneous cubic diophantine equations (1) x | y 3 + 1 and y | x 3 + 1, where x and y are positive integers. The main result in this paper is that there exist an infinite number of sequences such that x and y satisfy (1) if and only if they are consecutive terms of one of these sequences.