Thomas Bloom spends part of his life keeping score for a dead genius. He runs erdosproblems.com, the running ledger of questions the Hungarian mathematician Paul Erdős left unsolved when he died in 1996. So when OpenAI declared a mathematical triumph last October, Bloom was exactly the wrong person to bluff.
He called the company's claim "a dramatic misrepresentation."
This week, Bloom's name is on a paper vouching for OpenAI.
On May 20, OpenAI said one of its general-purpose reasoning models had produced an original proof disproving the planar unit distance conjecture, a question Erdős first posed in 1946. The company called it "the first time AI has autonomously solved a prominent open problem central to a field of mathematics." The difference between this announcement and the last one is not the confidence. It is that the mathematicians who humiliated OpenAI seven months ago checked the work first.
The October Fiasco Set the Bar for Doubt
To understand why this week matters, rewind to the embarrassment.
In October 2025, OpenAI's then vice president Kevin Weil posted on X that GPT-5 had "found solutions to 10 (!) previously unsolved Erdős problems and made progress on 11 others." It was the kind of line that travels fast. It was also wrong.
GPT-5 had not solved anything. It had located solutions that already existed in the mathematical literature and presented them as fresh. Bloom, whose website was the source being misread, said so bluntly. Yann LeCun and Google DeepMind chief Demis Hassabis piled on. Weil deleted the post.
That history is the reason OpenAI's May announcement arrived wrapped in external review rather than a victory lap. The company published companion remarks from working mathematicians alongside the result, instead of asking the public to take its word.
The Problem Is Simple to State and Brutal to Solve
The unit distance problem sounds like a children's puzzle. Place dots on a flat sheet of paper. Count how many pairs of dots sit exactly one unit apart. The question: as you add more points, how fast can that count of unit-distance pairs grow?
For nearly 80 years, the consensus answer pointed at the square grid. Arrange points in a tidy lattice and you get a predictable number of one-unit pairs. Erdős proposed that the count could only grow slightly faster than the number of points itself, never dramatically more. Generations of mathematicians tried and failed to settle whether anything could beat the grid.
OpenAI's model found something that does. According to the company, it discovered an infinite family of point arrangements that produce significantly more unit-distance pairs than the classic grid, breaking the picture mathematicians had carried for decades.
What unsettled experts was the route the model took. Instead of leaning on the usual tricks of combinatorial geometry, it reached into algebraic number theory, the study of exotic number systems that extend the ordinary integers. The proof used machinery rarely seen near this kind of geometry problem, including infinite class field towers and Golod-Shafarevich theory. In plain terms, the model used hidden symmetries buried inside strange number systems to manufacture far more one-unit distances than anyone expected was possible.
Princeton mathematician Will Sawin later refined the result, pinning the improvement to a fixed exponent rather than a vanishing one. The model found the door. A human helped measure how far it opened.
The Verification Chain Is the Real News
A bold claim from an AI lab is cheap. The names attached to this one are not.
The proof went through external review, and the reviewers produced a companion paper explaining the argument and why it matters. The roster reads like a who's who of the field:
- Tim Gowers, the Fields Medal winner, who called the achievement "a milestone in AI mathematics."
- Noga Alon and Melanie Wood, both leading figures in combinatorics and number theory.
- Arul Shankar, a number theorist who said the work shows AI systems can move past assisting mathematicians and start generating genuinely original ideas.
- Thomas Bloom, the same researcher who called OpenAI's October post a dramatic misrepresentation.
When the people who exposed your last mistake agree to put their reputations behind your next claim, that is the verification that counts. The October failure, painful as it was, is part of why this result is credible: the skeptics were already watching, and this time they did not flinch.
OpenAI's framing is that the model held together a long, difficult chain of reasoning and connected ideas across fields in ways researchers had not explored. That is the capability practitioners care about, far more than the geometry itself. A model that can sustain a multi-step argument and import tools from an unrelated branch of math is a model that might do the same in biology, physics, or engineering. For a working sense of why long, structured reasoning is the frontier worth watching, see our explainer on reasoning models and how AI learned to think step by step.
What the Proof Does Not Prove
The result is genuine. The hype around it should still be handled with tongs.
This is one problem. Disproving a single conjecture, however famous, is not the same as a machine independently building new mathematical theory at scale. The model found a construction; a human mathematician, Will Sawin, sharpened it into its cleaner final form. The line between autonomous discovery and very fast assistance is blurrier than a press release suggests.
There is also the matter of what "general-purpose" earns OpenAI. The company stresses the proof came from a reasoning model not built for math and not pointed at this problem in particular. That is the impressive part. It is also the unverifiable part, since outside researchers cannot audit how the model was prompted, how many attempts it took, or how much scaffolding sat around it. The mathematics has been checked. The process has not.
Skeptics like LeCun have spent years arguing that today's models pattern-match rather than reason. A clean, reviewed proof of an 80-year-old problem is a real data point against that view. It is not the end of the argument, and it sits alongside benchmarks that still expose how much these systems do not know, like the brutal evaluation we covered in Humanity's Last Exam.
The Bottom Line
Seven months ago, OpenAI mistook a library lookup for a discovery and got caught by the man who keeps the records. This week, that same man helped certify that one of OpenAI's models did something humans had not managed in 80 years: it broke the unit distance conjecture, and it did so by dragging deep number theory into a geometry problem nobody thought needed it.
The lesson for engineers is not that math is solved. It is that the gap between "the model sounds confident" and "the model is correct" now has referees, and on this problem the referees said yes. Whether the same model can do this on demand, on the next problem, without a Princeton professor to tidy up, is the question that actually decides how much this changes the work.
Bloom, for his part, sounded less like a skeptic than a kid who just found a trapdoor. "AI is helping us to more fully explore the cathedral of mathematics we have built over the centuries," he said. "What other unseen wonders are waiting in the wings?"
Sources
- An OpenAI model has disproved a central conjecture in discrete geometry — OpenAI, May 20, 2026
- OpenAI claims it solved an 80-year-old math problem — for real this time — TechCrunch, May 20, 2026
- 80-year-old geometry mystery cracked by OpenAI using deep number theory — Interesting Engineering, May 20, 2026
- Remarks on the disproof of the unit distance conjecture (companion paper) — arXiv, May 2026
- OpenAI's embarrassing math (October claim) — TechCrunch, October 19, 2025
- OpenAI Says It Solved an Erdős Puzzle, This Time For Real — Technology.org, May 21, 2026