# Erdős Problem 659

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:659
- Collection: Erdős
- Page: https://qed.bot/s/erdos-659
- Formal record: solved; at its source: proved (Lean)
- Machine-checked: no independent check has verified it
- Fidelity F0: Absent. No formal statement is attached to the result.
- Source: https://www.erdosproblems.com/659

## AI claims

### Aletheia, 2 Feb 2026

Full solution

- Outcome: full
- Autonomy A3: Autonomous. The system produced the result without significant human mathematical involvement.
- Evidence V1: Informal. A public write-up exists and has had community scrutiny.
- Fidelity F0: Absent. No formal statement is attached to the result.
- Source: Erdős problems AI wiki, https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems

### Gemini 3, 13 Jan 2026

Full solution

- Outcome: full
- Autonomy A1: Collaborative. A human and a system worked the problem together, or the system played a supporting role.
- Evidence V1: Informal. A public write-up exists and has had community scrutiny.
- Fidelity F0: Absent. No formal statement is attached to the result.
- With: Benjamin Grayzel
- Source: Erdős problems AI wiki, https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems

### Gemini Deep Research, 13 Jan 2026

Partial results, upgradeable to full solution, found

- Outcome: partial (a supporting task)
- Autonomy A1: Collaborative. A human and a system worked the problem together, or the system played a supporting role.
- Evidence V1: Informal. A public write-up exists and has had community scrutiny.
- Fidelity F0: Absent. No formal statement is attached to the result.
- Source: Erdős problems AI wiki, https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems

### Aristotle, 14 Jan 2026

Gemini and Grayzel (2026)

- Outcome: full (a supporting task)
- Autonomy A1: Collaborative. A human and a system worked the problem together, or the system played a supporting role.
- Evidence V1: Informal. A public write-up exists and has had community scrutiny.
- Fidelity F0: Absent. No formal statement is attached to the result.
- Source: Erdős problems AI wiki, https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems

## Formal statements

- FormalConjectures/ErdosProblems/659.lean, https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/659.lean

Cite as: qed.bot, "Erdős Problem 659", https://qed.bot/s/erdos-659, as of 30 Sep 2026. The register's data is published under CC BY 4.0: https://creativecommons.org/licenses/by/4.0/