HEXAGONbeta

Long solutions to small equations

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

Contributed by Bogdan Grechuk ↗, Eugene Go ↗, Avraham Eisenberg ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

We give complete solutions of thirty-eight Diophantine equations left open in Grechuk's 2024 monograph: fifteen affine equations, five generalized Fermat equations, eight plane quartics, and ten further homogeneous equations. All proofs were generated by AI and require substantial computer algebra computations.

Provenance statement

All proofs were generated by AI and require substantial computer algebra computations. Given the length of the arguments and the amount of supporting computation, it is difficult for a human to check every detail of the proofs. However, we understand the overall strategies of the arguments.

Tools used

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

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Version history

  1. v1Submitted by Bogdan GrechukInitial depositCurrentOct 24, 2026