Peer reviewed
Erdős #90 unit-distance conjecture disproved by an OpenAI model
Erdős #90 unit-distance conjecture disproved by an internal OpenAI model
The gist
An internal OpenAI model disproved Erdős's unit-distance conjecture (problem #90), and nine mathematicians published a verified version on May 20, 2026.
DiscoverIA status on October 11, 2026: peer reviewed, meaning published in a journal or reviewed by named experts.
Text written by an AI, reviewed by Antoine Olivier,
Cite this entry
A single status snapshot since Oct 11, 2026: a citation always points to a dated snapshot.
Cite snapshot 1, Oct 11, 2026
APA citation with the dated link
BibTeX @misc, key discoveria-erdos-90-unit-distance-r1
@misc{discoveria-erdos-90-unit-distance-r1,
author = {{DiscoverIA}},
title = {Erdős #90 unit-distance conjecture disproved by an internal OpenAI model},
howpublished = {DiscoverIA registry, entry erdos-90-unit-distance},
note = {Status “Peer reviewed”, snapshot 1, recorded on October 11, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/erdos-90-unit-distance#releve-1},
urldate = {2026-10-11}
}Dated link opens this snapshot in the history
To understand ≈ 20 s reading time, estimated
What it changes
To disprove a conjecture is to prove that it does not hold. This discrete geometry question posed by Erdős, about how many pairs of points can be exactly the same distance apart, therefore has a negative answer, and the review by named mathematicians gives the result firmer support than the announcement alone.
What remains unclear
The documented check covers a short, simplified version of the counterexample, published by these reviewers. No publication in a peer-reviewed journal has been found.
In the glossary : Conjecture
Read the paper
Remarks on the disproof of the unit distance conjecture
What it establishes : The paper gives a short, simplified proof of the counterexample: there are point sets in the plane where, for n points, the number of unit distances is at least n to the power 1 + ε (for a fixed ε > 0), built using number fields.
Where to start : The abstract, then Theorem 1.1 (Section 1); the sketch in Section 1.2 needs no background, and the full proof is in Section 2 (Lemmas 2.1 and 2.2). Sections 3 to 11 are each author's remarks.
Written by an AI · reviewed by a human
For experts
Sources, method, key facts, status history, report an error
Status
Sources
- Main sourcearxiv.orgpage read word for word
- Corroborationerdosproblems.compage read word for word ·
disproved by an internal model at OpenAI
Method
Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood publish a "digested, human-verified" version on arXiv (2605.20695); the sources read do not describe how the counterexample is built. OpenAI's original text was not reopened for this record. DiscoverIA status: peer reviewed (published in a journal or reviewed by named experts). Set on October 11, 2026 from arxiv.org (page read, quotation checked word for word: “short, digested, human-verified version of the recent OpenAI-generated counterexample”).
Limits
The status may change if a new source is published; the dated history stays visible.
Key facts
- Claimed by
- OpenAI (internal model)
announced on May 20, 2026 - Nature
- new result
- Field
- Mathematics (discrete geometry / combinatorics)
- Checked by
- 9 named mathematicians (simplified “human-verified” version)
- Sources
- 2 : 2 read word for word
- This entry
- Reviewed by Antoine Olivier, Oct 13, 2026
Status history
Only one observation so far.
- 01
First observation:Peer reviewedCurrent status
review by experts named in a preprint; no peer-reviewed journal publication found
Source: arxiv.orgread word for word
An observation is added; none is ever erased.
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