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Polynomial parametrizations for two classes of Diophantine equations

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Bogdan Grechuk ↗

Version 1 / Oct 27, 2026 / CC BY 4.0

Abstract

We prove that the integer solution sets of two classes of Diophantine equations are finite unions of polynomial families with integer coefficients and unrestricted integer parameters. In the first class, each monomial, with at most one exception, contains a variable that occurs to the first power and in no other monomial. The second class consists of equations M=f(t)M=f(t), where MM is a monomial independent of tt with a variable occurring to the first power, and f∈Z[t]f\in\Z[t] splits completely over $\Q$. If MM involves at most two variables, ff may instead be an arbitrary quadratic polynomial. We also give complete formulas for five equations listed as open in the monograph \emph{Polynomial Diophantine Equations: A Systematic Approach} (2024).

Provenance statement

This work was developed through an iterative collaboration between the contributor and OpenAI’s ChatGPT/Codex. The contributor selected the problems, determined the scope, and directed revisions. ChatGPT/Codex was used extensively to develop proof arguments and explicit parametrizations, search and check the literature, draft and revise the manuscript, and prepare the LaTeX source. A Claude Opus 5.5 review verified the argument and informed a further revision.

Tools used

OpenAI
ChatGPTVersion GPT-6 Astra Pro, GPT-5.6 Sol Ultra
Anthropic
Claude OpusVersion 5.5

References

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  8. B. Grechuk, Polynomial Diophantine Equations: A Systematic Approach , Springer, 2024. https://doi.org/10.1007/978-3-031-62949-5.DOI
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  10. B. Grechuk, A systematic approach to Diophantine equations: open problems , arXiv:2404.08518v10, 1 October 2026. https://arxiv.org/abs/2404.08518v10.arXiv

Version history

  1. v1 · 19 pagesSubmitted by Bogdan GrechukInitial depositCurrentOct 26, 2026