Diophantine equations with three monomials

TL;DR

This paper introduces a comprehensive algorithm for solving all two-variable polynomial Diophantine equations with three monomials, advancing the understanding of their solutions and classifications.

Contribution

It provides a general algorithm for these equations and an elementary reduction method, identifying a class of equations with complete solutions.

Findings

The algorithm solves all two-variable three-monomial equations.

A reduction method simplifies the solution process.

Empirical data indicates the class covers nearly all such equations as variables increase.

Abstract

We present a general algorithm for solving all two-variable polynomial Diophantine equations consisting of three monomials. Before this work, even the existence of an algorithm for solving the one-parameter family of equations $x^4+axy+y^3=0$ has been an open question. We also present an elementary method that reduces the task of finding all integer solutions to a general three-monomial equation to the task of finding primitive solutions to equations with three monomials in disjoint variables. We identify a large class of three-monomial equations for which this method leads to a complete solution. Empirical data suggests that this class contains $100\%$ of three-monomial equations as the number of variables goes to infinity.