The Integer Solutions to an Exponential Diophantine Equation in Three Variables

Abstract

The subspace theorem due to W. M. Schmidt implies that the unit equation in three variables has only finitely many solutions except trivial cases. However, there is no effective method in general to determine the solutions, hence very few examples of explicit solutions are known. In this article, we give a complete list of the integer solutions to an exponential Diophantine equation in three variables, which is indeed one unit equation in three variables.