On the diophantine equation $ax^{3} + by + c = xyz$

Abstract

Consider the diophantine equation $ax^{3} + by + c = xyz$, where $a, b$ and $c$ are positive integers such that $\gcd(a, c) = 1$ and $c$ is square-free. Let $(x, y, z)$ be a positive integral solution of the equation. In this paper, we shall give an upper bound for $x$,$y$ and $z$ in terms of the given inputs $a$, $b$ and $c$. Also, we apply our results to investigate the divisors of the elements of the sequence $\{an^{3} + c\}$ in residue classes.