OpenAI says model solved Erdős problem

OpenAI said on May 20 that one of its unreleased reasoning models generated a proof on the planar unit distance problem, a combinatorial geometry question first posed by Paul Erdős in 1946. The company said the model disproved a central conjecture tied to the problem and published the proof and companion remarks on its website. The claim was picked up in media reports on May 22, including The Indian Express, which said OpenAI described the result as an original mathematical proof on a question that had remained open for nearly 80 years. Paul Erdős’s question asks how many pairs of points in a plane can be exactly one unit apart when there are \(n\) points in total. OpenAI said that problem became one of the best-known questions in combinatorial geometry because it is easy to state but difficult to resolve. The company’s post said the new proof overturns a major conjecture about the best possible construction for maximizing those unit-distance pairs. (openai.com) ### Which math problem does OpenAI say its model solved? The May 20 OpenAI post identified the target as the planar unit distance problem, sometimes called the Erdős unit distance problem. OpenAI said mathematicians have studied the question for decades and that a 2005 research text listed it among unsolved problems in discrete geometry. The company said its model found a proof that disproves a conjecture that square-grid constructions are asymptotically optimal. (openai.com) Scientific American reported on May 22 that the result concerns the 80-year-old “unit distance” conjecture and described it as a proof that, if produced by humans alone, would likely be publishable in a top mathematics journal. That assessment was attributed by the magazine to mathematicians it interviewed, not to OpenAI. (openai.com) ### What exactly did OpenAI claim the model produced? OpenAI said the model produced an original proof rather than retrieving an existing one from the literature. The company published both the proof and a separate set of remarks alongside the announcement. The Indian Express reported that OpenAI said the model was unreleased and designed for reasoning tasks. The OpenAI post framed the result as a research milestone in AI-driven mathematics. (scientificamerican.com) The company did not say in the post surfaced by search that the model itself had been publicly released or added to ChatGPT. ### Have outside mathematicians weighed in? Scientific American reported that mathematicians were “amazed” by the result and said the proof could meet the standard of a leading journal if humans had produced it alone. (openai.com) That article described the work as the first AI proof of that likely caliber. The report does not amount to formal journal acceptance, but it is the clearest external characterization surfaced in current coverage. The Indian Express report said OpenAI’s proof had been checked by external mathematicians, citing the company’s account of the process. OpenAI’s own post made the proof available for review, which allows researchers to inspect the argument directly. ### Why is Daybreak showing up in the same news cycle? (scientificamerican.com) OpenAI’s Daybreak webpage says the company is using GPT-5.5 and Codex Security in a cybersecurity product focused on threat identification, patch generation and remediation verification. The page describes Daybreak as an offering for cyber defense workflows across code and systems. An OpenAI post published two weeks earlier said GPT-5.5 was already delivering cybersecurity capabilities through Trusted Access for Cyber. (indianexpress.com) A separate OpenAI product post on April 23 introduced GPT-5.5 as the company’s latest model family and said it was rolling out in ChatGPT, Codex and the API. ### What comes next for the math claim? OpenAI has already posted the proof and companion remarks on its site, making the next step scrutiny by mathematicians rather than another product launch. (openai.com) Scientific American’s report points to possible journal-level interest, while current OpenAI materials do not announce a publication decision or release date for the unreleased reasoning model. (openai.com 1) (openai.com 2)