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Almost all kk-Markov numbers are unique

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.DS — Dynamical Systems

Contributed by Qiyue Tang ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

For every fixed nonnegative integer kk, we prove that almost every positive integral solution of [x^2+y^2+z^2+k(xy+xz+yz)= 3(k+1)xyz]is uniquely determined, up to permutation, by its largest coordinate.

Provenance statement

GPT-6-Pro was used in one research run lasting approximately 3.2 hours (3 hours and 12 minutes). GPT-6-Pro was also used for subsequent mathematical verification, adversarial review, and manuscript polishing. An additional independent review by Rethlas-OSV is planned and will be incorporated later.

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Version history

  1. v1 · 22 pagesSubmitted by Qiyue TangInitial depositCurrentNov 08, 2026