- An AI produced an original mathematical proof, not just a literature search
- It disproved Erdős's 1946 unit-distance conjecture by using higher dimensions
- About 100 Erdős problems are now 'solved' with AI help since October 2025
The American mathematician Mehtaab Sawhney was working through Paul Erdős's old conjectures the way mathematicians have for decades, problem by stubborn problem. But Sawhney had a new weapon in his tool belt: he has been employing AI in mathematics work.
A few months earlier, Sawhney had taken leave from Columbia to join OpenAI. His colleague Mark Sellke, recently on leave from Harvard, joined him in feeding an 80-year-old geometry problem to one of the company's internal reasoning models.
The model's chain of reasoning stretched to hundreds of pages.
When it stopped, it had done something most experts believed current AI could not do. It produced a genuine, original proof. The proof overturned a long-standing conjecture by Erdős, the eccentric Hungarian mathematician who spent his career posing puzzles designed to resist the best human minds.
Paul Erdös (1913-1996) was a famous Hungarian mathematician who produced a large number of mathematical conjectures in the 20th century. Photo by Kmhkmh - CC BY 3.0,
"It feels like magic," Sawhney told Scientific American. "It's kind of an amazing experience to have a machine give back something which really resembles how I work."
The result was announced by OpenAI in May 2026. Experts say it is the first AI-generated proof that would merit publication in a top mathematics journal if humans had done it alone.
An 80-year-old trap dressed as a geometry puzzle
The unit-distance problem sounds like something a curious child could ask.
What is it?
The unit-distance problem
Place any number of dots on a flat surface. How many pairs of dots can be exactly one unit apart from each other? Posed by Paul Erdős in 1946, the problem asks for the maximum possible count as the number of dots grows. For 80 years, mathematicians believed a carefully spaced grid was essentially optimal.
Place some dots on a flat piece of paper. How many pairs of those dots can be exactly one unit apart? With nine dots in a row, you get eight such pairs. Arrange them in a three-by-three grid and you get twelve.
For any number of dots, what is the highest possible count?
In 1946, Erdős guessed at the answer. A grid, with the spacing chosen carefully using deep number theory, could squeeze out slightly more pairs than a simple lattice. He conjectured that nothing could do meaningfully better than this.
For 80 years, no one disproved him.
"For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids," OpenAI wrote in its announcement. That collective faith, characteristically Erdősian in how long it survived without proof, turned out to be the conjecture's protective shell.
Example of previously known construction of many unit distances from a rescaled square grid. Adapted from openai.com.
A shadow from a higher dimension
The model did not invent a new branch of mathematics. It just went up to a higher dimension.
Instead of placing dots on a flat page, it constructed an elaborate lattice in higher-dimensional space. The lattice had mathematical symmetries that pack pairs of points at the same distance far more efficiently than any planar grid.
The model then found a way to project that strange shape back down to two dimensions. The result was what Sawhney described as a numerical "shadow" of the higher-dimensional construction.
You can read the model's reasoning in a separate document - it is a fascinating read.
"What the model did is totally different from the 'square grid' construction," Sellke said.
The proof improves on the Erdős bound by a polynomial factor. Princeton mathematician Will Sawin, working from the model's output, has since sharpened the gain to an exponent of 0.014. That sounds small, but represents an infinite family of arrangements better than anything previously known.
The flattened shadow, Sawhney noted, is too tangled to draw on paper even for modest numbers of dots.
The patience humans couldn't afford
The question is not whether the AI did real mathematics. (It did.) Rather, it is why no human had walked this particular path.
The tools were sitting in the literature. Sébastien Bubeck, the mathematician leading OpenAI's mathematical work, put the matter plainly. "The model did not invent something fundamentally new that nobody saw coming. It just executed like an amazing mathematician."
What the model had, that humans lacked, was time and a peculiar indifference to discouragement.
Most mathematicians who attacked the unit-distance problem were trying to prove Erdős right, not refute him. The few who looked for counterexamples were unlikely to pursue a difficult, high-dimensional construction without some early hint of success.
The grind was tedious, and the payoff uncertain, which made it a high-cost, low probability venture. Human researchers had careers, students, other open problems to consider.
AIs have an edge. It's not just that they can try all known methods. They can play for longer and in more treacherous waters than mathematicians without getting overwhelmed.
Jacob Tsimerman, University of Toronto
"AIs have an edge," Jacob Tsimerman, a mathematician at the University of Toronto, told Scientific American (paywalled). "It's not just that they can try all known methods. They can play for longer and in more treacherous waters than mathematicians without getting overwhelmed."
Harvard mathematician Melanie Matchett Wood, who reviewed the proof for OpenAI, put it more bluntly. Experts had spent considerable effort parsing the AI's answer after the fact. If they had spent the same effort hunting for a counterexample beforehand, she suggested, they would have found one themselves.
"Maybe people should be spending more time, you know, playing devil's advocate."
October's broken claim, May's verified proof
This story has a useful predecessor.
In October 2025, OpenAI's then-vice-president Kevin Weil announced on X that GPT-5 had solved ten Erdős problems. Within days, Thomas Bloom, the mathematician who maintains the official Erdős problems website, pointed out the awkward reality.
GPT-5 had not produced original proofs. It had found existing solutions, sitting unread in old papers, that no one had connected to Erdős's list. The result was useful. It was also nothing like what the headlines said.
OpenAI handled the May 2026 result differently. Before announcing anything, the company quietly sent the proof to a panel of external mathematicians for review. The list included Daniel Litt at Toronto, Fields medalist Tim Gowers at Cambridge, Sawin, Wood, and Bloom himself.
The reviewers wrote up their reactions in a companion document. Gowers called the result "a milestone in AI mathematics."
Litt, who is not affiliated with OpenAI, was harder to impress and still struck. "This is the unique interesting result produced autonomously by AI so far," he said.
Where AI in mathematics stands in 2026
The unit-distance proof sits inside a broader transformation that mathematicians are still trying to take the measure of.
Since October 2025, according to a tracker started by Terence Tao, AI tools have helped move roughly 100 Erdős problems into the "solved" column. The bulk of that work has been souped-up literature search, the kind that found Bloom's missed references.
Erdős problems moved with AI help
~100
Problems shifted to "solved" since October 2025, per Terence Tao's tracker. Most via literature search; a few via genuine original proofs.
Some has involved language models piecing together existing theorems into new proofs, in dialogue with human collaborators. In at least two cases, an AI built an original proof of an Erdős problem with little human input.
The change has been quick enough that mathematicians are reorganising their careers around it. Sawhney took leave from Columbia. Sellke did the same.
Carlo Pagano, who has collaborated with Google DeepMind on Erdős problems using Gemini, recently started a joint position at the company. "It's clear that this will change how we do math," Pagano said earlier this year, "so better to start early rather than later."
Critical discussions on the proof
The story's coverage in the media shows that AI in mathematics is getting a lot of attention, but not all agree that the "AI finds mathematical proof" angle is quite right. Is AI really figuring this out on its own, or is the tool simply doing what it was programmed for?
Carlo Pagano himself has called the Erdős problems imperfect benchmarks, useful precisely because they are numerous and overlooked rather than because they are central. No major mathematics journal has yet published a peer-reviewed proof crediting AI as a contributor.
Cal Newport, a computer scientist at Georgetown, raised a different question in remarks Gary Marcus published the day after the announcement. What if the result says less about machine intelligence than about which problems happen to suit current machine architecture?
Mathematics and coding are the two domains where reasoning models perform best. They are also two of the smaller markets a company like OpenAI could pursue.
More on AI in Mathematics
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→Newport offered a sharper analogy. The model is not a better mathematician than its human reviewers, he argued. It is closer to what CAD software does for architects: making bolder designs feasible, not replacing the designer.
Marcus added a quieter caveat. OpenAI shared the proof but not the failures. We have a numerator without a denominator, and no public record of how many prompts produced nothing.
The AI's quieter shortcomings remain. Wood noted that the model presented several ideas as its own when very similar ones already existed in the literature.
"If a human had been familiar with those results and not credited them, then that would be professional malpractice."
AI is changing mathematics
Wood, asked what mathematicians should take from all of this, did not offer a forecast or a warning.
"Any mathematician who hasn't been using the latest models should be surprised," she said. "It's quite a different world than in December of last year."
Mathematicians may now need to count on AI.
Sources
- Primary source: "An OpenAI model has disproved a central conjecture in discrete geometry." OpenAI, May 20, 2026
- Context sources:
- Howlett, Joseph. "AI just solved an 80-year-old 'Erdős problem,' and mathematicians are amazed." Scientific American, May 21, 2026.
- Howlett, Joseph. "Is AI on the precipice of revolutionizing math? It depends." Scientific American, February 12, 2026.
- "OpenAI Says It Solved an Erdős Puzzle, This Time For Real." Technology.org, May 2026.
- "OpenAI Finally Solved a Real Math Problem." AutoGPT, May 2026.
- Tao, Terence. AI contributions to Erdős problems (tracker).
- Sawin, Will. Improved bounds for the unit-distance problem. arXiv:2605.20579.
Fact Check: Claim-by-Claim Verification Verified
All major factual claims about OpenAI's May 2026 disproof of Erdős's unit-distance conjecture verified across three rounds of dialogue with Perplexity Sonar Pro. Key affiliations, quotes, and the historical October 2025 GPT-5 episode all check out against primary sources.
Editor note: Round 2 Perplexity flagged this as "Columbia, not Princeton" based on outdated Clay Math Institute and Columbia news pages, but Round 3 confirmed via Sawin's own homepage that he is currently at Princeton. No fix needed.
Commentary
- The "almost no one seriously tried" framing about the 80 years before this disproof is rhetorical compression — there was real (if incremental) work on the problem.
- The Bubeck quote could not be directly verified against a primary source; the sentiment matches his public role but exact wording should be considered paraphrase-grade.
- Carlo Pagano's role at DeepMind is described as a "joint position"; reporting confirms a position but the joint-with-Concordia structure is plausible inference, not directly cited.
- The "external review panel" framing simplifies a broader set of nine coauthors on the companion arXiv paper. Article names a subset, which is accurate but partial.
Sources used for verification
Academic / Peer-reviewed:
- An explicit lower bound for the unit distance problem — arXiv (Sawin)
- Remarks on the disproof of the unit distance conjecture — arXiv companion paper (Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang, Wood)
- Paul Erdős biography — MacTutor
Other reliable sources:
- AI just solved an 80-year-old Erdős problem — Scientific American
- Mathematicians stunned by AI's biggest math breakthrough yet — New Scientist
- An OpenAI model has disproved a central conjecture in discrete geometry — OpenAI
- Checking the math behind OpenAI and Anthropic's latest headlines — Gary Marcus on Substack
- AI contributions to Erdős problems — Terence Tao's GitHub wiki
- Erdős Problems website — Thomas Bloom (Manchester)
- Will Sawin homepage — confirms current Princeton affiliation
- OpenAI Finally Solved a Real Math Problem — AutoGPT (October 2025 GPT-5 context)
Fact-checked by Perplexity Sonar Pro on 2026-05-23
