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arXiv · 2307.02513

Diophantine equations with three monomials

Bogdan Grechuk·Tetiana Grechuk·Ashleigh Wilcox
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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.

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Bogdan Grechuk, Tetiana Grechuk, Ashleigh Wilcox. 2023-07-05. Diophantine equations with three monomials. https://arxiv.org/abs/2307.02513

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