Breakthrough Tracker record

A preprint claims a full proof of Fraenkel’s Beatty-sequence conjecture

Hu Tan and Ying Zhang claim that every exact partition of the integers into at least three Beatty sequences with distinct moduli has Fraenkel’s binary density pattern. Their proof reduces the problem to a dimension-free one-third density theorem, exact finite checks and a deletion induction.

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Stable ID: math-fraenkel-beatty-sequence-conjecture-proof-2026
Revision: math-fraenkel-beatty-sequence-conjecture-proof-2026.v1
Field: Mathematics · Number theory, combinatorics and discrete Fourier analysis
Evidence: Tier 1 · Peer reviewed: No
Record state: Provisional · Provisional claimed proof
Last checked: 2026-09-04

AI role

The authors report that GPT-5.6 Sol resolved the cases with 8–11 components and inspired key proof ideas; OpenAI Codex designed, implemented and ran the finite-case verification code; and Ziv supported exploration and checking. They state that the authors determined and verified the mathematics.

Record details

Problem or result: Fraenkel’s 1973 classification of exact partitions by distinct Beatty sequences
Authors: Hu Tan and Ying Zhang
Institutions: Academy of Mathematics and Systems Science, Chinese Academy of Sciences; Soochow University
Result date: First-version preprint submitted September 1, 2026

Why it matters

The conjecture links exact integer coverings to a rigid binary pattern and had previously been proved only through seven components. A correct general proof would close the remaining cases uniformly.

Limits

This is a 36-page first-version preprint with no located independent correctness assessment or peer review. The exact finite checks have public code, but the announced Lean formalization was still in progress when this record was checked.

Sources

Primary: Tan and Zhang, claimed proof of Fraenkel’s conjecture
Artifacts: Finite-case verification repository

Correction and revision history

2026-09-04 — Added after primary-source, scope, status, AI-role and limitation review.

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