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Diophantine equations with three monomials

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journal contribution
posted on 2023-12-14, 11:17 authored by B Grechuk, T Grechuk, A Wilcox

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 x4+axy+y3=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. Video: For a video summary of this paper, please visit https://youtu.be/ANsJvFTmDCE.

History

Author affiliation

School of Computing and Mathematical Sciences, University of Leicester

Version

  • VoR (Version of Record)

Published in

Journal of Number Theory

Volume

253

Pagination

69 - 108

Publisher

Elsevier BV

issn

0022-314X

eissn

1096-1658

Copyright date

2023

Available date

2023-12-14

Language

en

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