AI Research• 11 min read

May 2026: An OpenAI Model Disproves an 80-Year-Old Erdős Conjecture

On May 20, OpenAI revealed that an internal general-purpose reasoning model had disproved the Erdős unit distance conjecture, arguably the best-known problem in discrete geometry, and nine outside mathematicians published a human-verified version of the proof the same day. Tim Gowers called it the first famous open problem solved by AI with no human intervention once the model was given the problem. It is a genuine milestone for AI in research, and its caveats about verification, credit and failed attempts matter almost as much as the headline.

Author: Macaulan Chiaramonte

May 2026 produced the clearest evidence yet that AI can contribute original results to research mathematics. An internal OpenAI model produced a counterexample to a conjecture Paul Erdős posed in 1946, showing that points in a plane can have far more pairs exactly one unit apart than Erdős believed possible. Mathematician Will Sawin made the gain explicit the same day: for some sets of n points, the number of such pairs exceeds n to the power 1.014.

The verification was as notable as the result. Nine mathematicians, among them Gowers, Noga Alon and Jacob Tsimerman, co-wrote a companion paper on arXiv, and Tsimerman wrote that he would accept the work "for any journal without hesitation." Others used the same paper to warn about what the headline leaves out: the attempts that fail, the difficulty of checking AI output, and the question of who gets credit.

The Problem: How Often Can One Distance Repeat?

The question is easy to state. Place n points on a plane: how many pairs can be exactly one unit apart? Alon's reflection quotes an influential collection of research problems that calls it "possibly the best known (and simplest to explain) problem in combinatorial geometry," and he adds that every mathematician working in that area has thought about it.

Erdős showed that a square grid achieves only slightly more pairs than points, and conjectured that nothing could do substantially better. The best proven upper bound, n to the power 4/3, dates to Spencer, Szemerédi and Trotter in 1984. For four decades the gap between the two stood, and most experts believed Erdős was right.

The problem carried real prestige. Erdős offered $300 for a proof or disproof in 1982 and later $500, so Thomas Bloom, who runs the erdosproblems.com database where it is Problem #90, writes that "we can unambiguously say that the AI has solved a $500 Erdős problem."

On April 16, Bloom had put the problem on a "Top 10 Erdős Problems" list, written to show that many Erdős problems are far from trivial. His reaction in the companion paper is telling: "I did not expect this to happen just one month later!"

What the Model Found

According to Scientific American, an OpenAI team whose members had already used AI on less prestigious Erdős problems fed this one to an internal model trained for general reasoning and asked whether Erdős was right. After hundreds of pages of logic and calculation, it beat his record. The verifiers write that the proof was "first mathematically generated in one shot" by the model, then refined for exposition through human interactions with Codex.

The construction generalizes Erdős's own. Where he used a square grid, the model built "a more elaborate grid, one living in a kind of higher dimension," in Scientific American's description, and mapped it back down to the plane. Technically, it works with number fields of ever-larger degree drawn from infinite class field towers, using Golod-Shafarevich theory from algebraic number theory rather than tools from geometry.

The model's reasoning trace, which the paper quotes, shows where the idea came from. "Maybe that enormous degree is not just an annoyance but a source of possible counterexamples," it reasoned. "Number fields deserve a closer look." Arul Shankar notes that a significant majority of the model's thoughts went into constructing a counterexample to the widely believed bound rather than trying to prove it.

That choice mattered. Shankar reads it as a mix of good intuition, a "willingness to try approaches considered long-shot by the community" and a predisposition to attempt constructions. When the model reached the crucial idea, he writes, it "honed in on the proof quite methodically."

How the Proof Was Checked

OpenAI privately contacted outside mathematicians to verify the proof. Daniel Litt of the University of Toronto writes that Mark Sellke and Mehtaab Sawhney at OpenAI asked him to check it because of his familiarity with the number theory involved, and that "it did not take long" to convince himself it was correct. According to Scientific American, the experts wrote up their takeaways without the company's direct involvement.

The result is a 19-page note, "Remarks on the disproof of the unit distance conjecture", presenting what the authors call "a short, digested, human-verified version" of the counterexample, followed by individual reflections. The same day, Sawin posted a separate paper making the exponent explicit at 1.014 and describing the OpenAI result as "drastically improving on the best previous lower bound."

Brevity helped. Victor Wang notes that because the digested argument fits in a few pages, at the level of a first- or second-semester graduate course in algebraic number theory, verification was relatively smooth. A longer argument, he suggests, might have found fewer volunteers willing to check the details.

There is an important limit on that verification. Scientific American reports that no external expert has seen the AI's original output, only an edited version of its train of thought. Bloom stresses that "the human still plays a vital role in discussing, digesting, and improving this proof, and exploring its consequences."

What the Mathematicians Said

The assessments were strong and specific. Alon called the solution "an outstanding achievement, settling a long-standing open problem," and added that "the AI was able to do here what lots of excellent human researchers tried and failed to do." Tsimerman, who says he once tried and failed to build a counterexample himself, would accept it for any journal.

Shankar went furthest. In his view, the paper shows that current AI models "go beyond just helpers to human mathematicians — they are capable of having original ingenious ideas, and then carrying them out to fruition." Litt called it "the first example of a result produced autonomously by an AI that I find exciting in itself, as opposed to as a leading indicator."

Gowers described a revealing first reaction. Hearing about the result on a call, he initially thought the model had proved Erdős's bound, and spent an evening "adjusting my world view." Learning it was a disproof "came as a big relief," because a counterexample was easier to imagine coming from a model that still lacks some essential mathematical capabilities.

Even so, Gowers wrote that this is the first example of a famous open problem solved by AI with no human intervention once the model was given it, and, in commentary quoted by Scientific American, that "no previous AI-generated proof has come close" to that standard. Gil Kalai compared it to the 1976 computer-based proof of the four color theorem, a possible landmark "whose importance goes beyond combinatorics and beyond mathematics."

Why AI Found What Experts Missed

The verifiers offer several explanations. Bloom notes that Erdős consistently believed his bound was true, so most human effort likely went into proving it rather than disproving it. Melanie Matchett Wood believes that if the same experts had been assembled a month earlier to hunt for a counterexample, they would have found one.

But without the AI's claim, Wood adds, nobody had a reason to assemble that group or point it at this problem. Her advice to colleagues, as quoted by Scientific American: "Maybe people should be spending more time, you know, playing devil's advocate."

Tsimerman points to stamina. The construction involves many uncomfortable choices, and in his words, AI systems "can play for longer and in more treacherous waters than mathematicians without getting overwhelmed." Litt offers another possible explanation: the solution needed ideas from a field most people working on the problem did not know, which would suggest that "specialization and silo-ing, though understandable, have cost us some high-quality science."

OpenAI's own researchers are measured about the nature of the achievement. "The model did not invent something fundamentally new that nobody saw coming," Sébastien Bubeck, who leads OpenAI's mathematical explorations, told Scientific American. "It just executed like an amazing mathematician."

Scale is part of the story too. As Understanding AI reported, an OpenAI chart showed that even at the maximum token budget, the internal model solved the problem only half the time. In this analysis, that is the more useful way to think about the capability: a search process good enough to be worth running repeatedly, rather than a single flash of insight.

The Caveats: Failed Attempts, Persuasion and Credit

Wood's reflection contains the most important caution in the paper. "This result does not show us all the times AI has claimed to have a proof of something and been wrong," she writes. Without that context, she argues, it is easy to draw incorrect conclusions about where AI and research mathematics stand.

Her sharper warning concerns verification itself: "In many cases, it will be easier for AI to convince humans it has a proof than to come up with a correct mathematical argument, and I believe that we as mathematicians are not sufficiently prepared for this." That concern applies well beyond mathematics, to any field where AI output is persuasive but expensive to check.

Credit is the other open question. Wood notes that closely related ideas in the literature are not appropriately referenced in the AI's write-up, and that citation norms built for human authors do not map cleanly onto a system that is in some sense familiar with all previous work. Wang adds that "the implicit social contract between mathematicians and AI companies deserves further attention."

Finally, the model itself is internal. Outside researchers could evaluate the digested proof, but not the system that produced it or its unedited output. Those limits do not undercut the result; they define how far it can be generalized.

How May Connects to the Year So Far

This blog's January recap covered Davos predictions that AI would reach "Nobel-level" research within two years, alongside skepticism that today's architectures could get there. May's result does not settle that argument, but it moves it from prediction to evidence: a verified, journal-quality result on a central problem, produced autonomously.

It also rhymes with April. Anthropic said Claude Mythos Preview's cyber skills were not explicitly trained but emerged from general improvements in code, reasoning and autonomy. OpenAI's result came from a model trained for general reasoning, not a mathematics system. In both cases, a frontier capability in a specialized domain arrived as a side effect of general progress.

The rest of the industry is moving the same way. At I/O the same week, Google introduced Gemini for Science, including Science Skills, a bundle that connects agents in its Antigravity platform to more than 30 life science databases and tools such as UniProt and the AlphaFold Database. Google says it lets researchers run workflows like genomic analyses in minutes rather than hours.

That pattern matters for planning. It suggests that breakthroughs in domains with checkable answers may come from general models rather than purpose-built ones, and on short notice. Alon, who had briefly experimented with ChatGPT Pro 5.5, wrote that the solution convinced him "it is hard to overestimate the full potential impact of this change."

What R&D and Technology Leaders Should Take From It

The lessons are practical, and most apply well outside mathematics:

  • Point models at backlogs: Litt notes the best autonomous AI math has come from "trawling through entire problem lists." Hard, checkable backlogs are a natural target.
  • Use AI as devil's advocate: Experts missed this partly because they believed the conjecture. Ask models to attack your domain's consensus assumptions.
  • Invest in verification: Expert checking time was the binding constraint. Plan reviewer capacity, and favor outputs short enough to audit.
  • Budget for repeated attempts: Success came about half the time at maximum budget. Treat inference compute as an R&D line item.
  • Track the misses: Wood's warning applies to every pilot. Measure failure and false-claim rates, not just the highlights.
  • Set provenance rules: Decide how AI-suggested ideas are attributed and how prior work is cited in your papers, patents and code.

Wood's counterfactual is the key point for managers. The experts could probably have found the answer, but nobody had a reason to look. Litt makes the same point from the other side: AI's sweep through whole problem lists is "a vast expansion of the attention aimed at mathematical problems," and may help focus where human attention goes next.

In this analysis, the organizations that gain most from this shift will be those that pair model capacity with genuine expert judgment. The model found the construction; nine mathematicians made it trustworthy.

Also in May

  • GPT-5.5 Instant (May 5): OpenAI made it ChatGPT's new default model, citing fewer hallucinations in sensitive areas such as law, medicine and finance.
  • Anthropic and SpaceX (May 6): Anthropic agreed to use all of the compute at SpaceX's Colossus 1 data center: more than 300 megawatts and over 220,000 NVIDIA GPUs.
  • Google I/O (May 19–20): Gemini 3.5 Flash became generally available and the new default in AI Mode, which Google says has passed 1 billion monthly users.
  • White House order pulled (May 21): President Trump delayed signing an order on voluntary pre-release testing of frontier models, saying it "could have been a blocker."

Looking Forward: From One Proof to a Research Pipeline

May showed that a general-purpose model can resolve a famous open problem and pass expert scrutiny. The open question is whether that becomes routine. Litt notes that short, clever solutions to well-known problems have historically been rare, but told Scientific American: "my guess is we're about to find out they're actually not that rare."

Key developments to watch in the coming months:

  • More Erdős problems: If Litt is right, more long-standing problems with short, clever answers will fall, starting with those on public lists.
  • Formal verification: Wang points to formalization alongside AI; machine-checked proofs would ease the human checking bottleneck.
  • Access to the model: The result came from an internal model; watch whether the capability reaches tools researchers can use.
  • Credit norms: Wood calls for shared standards on citation and attribution when AI suggests the ideas.
  • Other checkable domains: Watch for comparable results in algorithms, software and science, where answers can be verified.

May 2026 may be remembered as the month AI earned a place in the research literature on its own merits, with a result nine mathematicians were willing to put their names to. The deeper lesson is about workflow: machines that search tirelessly across fields, paired with experts who can judge what they find. Organizations that build that pairing now will be ready when the next result lands in their own domain.

Sources

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