# Erdős Problem 689

A statement in qed.bot, the register of claims of AI work in mathematics. The register grades the evidence attached to each claim; it never rules on whether a proof is correct.

- Identifier: erdos:689
- Collection: Erdős
- Page: https://qed.bot/s/erdos-689
- Formal record: open; at its source: open
- Machine-checked: yes, by Palomar
- Fidelity F3: The project checks its proof with Comparator against a statement from a corpus written separately from the proof, and held here.
- Source: https://www.erdosproblems.com/689

## AI claims

### Codex, GPT-5.2, GPT-5.5 Pro, 29 Oct 2025

Candidate full solution

- Outcome: candidate
- Autonomy A1: Collaborative. A human and a system worked the problem together, or the system played a supporting role.
- Evidence V3: Machine-checked. The artifact was rebuilt at a pinned commit by someone other than its authors — qed.bot, or a registry that publishes the check — and it holds: a proof that elaborates on the standard axioms alone, or an object that meets the problem's own constraints. Checked by Palomar.
- Fidelity F3: Anchored. The proof is checked for exact statement identity against a statement written independently of it, held here from a separate statement corpus.
- With: Boris Alexeev, Przemek Chojecki, Dogmachine, jleng01, Mehtaab Sawhney, Terence Tao, Malek Zribi
- Source: Erdős problems AI wiki, https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems

## Claimed on erdosproblems.com

Proof claims posted on erdosproblems.com, which says that listing a claim is no guarantee of proof correctness. Nobody has examined them, and none counts in the register's totals.

- Malek Zribi, Przemek Chojecki, using GPT 5.6 Sol Ultra, GPT 5.5 Pro, Claude Opus 4.8: a full proof claimed, 21 Jul 2026. Autonomy A?, evidence V0. https://www.erdosproblems.com/forum/thread/689/proof-claims#proof-claim-103

## Checks

- verified, by Palomar: Registered by Palomar at 54f27258: Comparator confirmed 1 theorems prove the recorded statement within Palomar's axiom policy, replayed through Lean's kernel and the independent nanoda kernel (https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-20-000002)

## Formal statements

- FormalConjectures/ErdosProblems/689.lean, https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/689.lean
- FormalConjectures/GreensOpenProblems/45.lean, https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/GreensOpenProblems/45.lean

Cite as: qed.bot, "Erdős Problem 689", https://qed.bot/s/erdos-689, as of 30 Sep 2026. The register's data is published under CC BY 4.0: https://creativecommons.org/licenses/by/4.0/