OpenAI Model Disproves Erdős Unit Distance Conjecture
An OpenAI model has achieved a major breakthrough in mathematics, by disproving a central conjecture in discrete geometry that was first posed by Paul Erdős in 1946.
This is the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
Today, we’re sharing that a general-purpose internal @openai model achieved a breakthrough on one of the best-known combinatorial geometry problems. Less than 1 year ago frontier AI models were at IMO gold-level performance. I expect this pace of progress to continue.
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
This is a general-purpose LLM. It wasn’t targeted at this problem or even at mathematics. Also, it’s not a scaffold. We have not pushed this model to the limit on open problems. Our focus is to get it out quickly so that everyone can use it for themselves.

Today, we’re sharing that a general-purpose internal @openai model achieved a breakthrough on one of the best-known combinatorial geometry problems. Less than 1 year ago frontier AI models were at IMO gold-level performance. I expect this pace of progress to continue.
Today, we’re sharing that a general-purpose internal @openai model achieved a breakthrough on one of the best-known combinatorial geometry problems. Less than 1 year ago frontier AI models were at IMO gold-level performance. I expect this pace of progress to continue.
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
Excellent thread from mathematician Tim Gowers on the significance of the @OpenAI model’s breakthrough on the Erdos Unit Distance Problem!
If you are a mathematician, then you may want to make sure you are sitting down before reading further.
@BorisMPower Congrats to the result, still disagreeing with the sentiment.
A general purpose model made this breakthrough at the heart of geometry. Exciting time ahead and probably no need for specialized models here!
Whatever the definition of "superhuman AI mathematician" is, I think my original prediction of June 2026 is not too far off the mark.
AI has now solved a major open problem -- one of the best known Erdos problems called the unit distance problem, one of Erdos's favourite questions and one that many mathematicians had tried. https://openai.com/index/model-disproves-discrete-geometry-conjecture/
@tunguz I am getting less and less confident about predicting. I think we have just entered the prediction event horizon where all bets are off.
@ChrSzegedy Yup, I think you nailed that prediction. What are some of your other predictions?
Once AI starts making solving open problems in novel ways it won’t stop.
We are entering the final stage of human solutions to open problems like this.
Feels weird, doesn’t it?
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
June 2024: The latest general-purpose LLMs could not count the r's in strawberry. July 2025: The latest general-purpose LLMs get gold in the International Math Olympiad. May 2026: The latest general-purpose LLM solve one of the "best-known questions in combinatorial geometry"
More on the solution: https://openai.com/index/model-disproves-discrete-geometry-conjecture/
June 2024: The latest general-purpose LLMs could not count the r's in strawberry. July 2025: The latest general-purpose LLMs get gold in the International Math Olympiad. May 2026: The latest general-purpose LLM solve one of the "best-known questions in combinatorial geometry"
The frontier is still jagged though (here is Gemini 3.5 Flash messing up counting letters in words)
Its The Graph again (not the METR graph, the one from the o1 launch).
Although no logarithmic decay of ability with increasing compute...
This is a general-purpose LLM. It wasn’t targeted at this problem or even at mathematics. Also, it’s not a scaffold. We have not pushed this model to the limit on open problems. Our focus is to get it out quickly so that everyone can use it for themselves.
Very striking, as is the linked post:
AI has now solved a major open problem -- one of the best known Erdos problems called the unit distance problem, one of Erdos's favourite questions and one that many mathematicians had tried. https://openai.com/index/model-disproves-discrete-geometry-conjecture/
Let's break this down, step by step [...]

Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
it’s kind of fucking ridiculous (and quite frightening) we‘re this far — the models are solving long standing problems in discrete geometry — yet the models do this still by thinking to themselves in plain english? that is easily interpretable? what the hell man

what a moment. wow. a bit in shock
what a moment. wow. a bit in shock
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
i’m so curious who on seb’s team just YOLOed planer unit distance into the latest checkpoint one night. doesn’t seem like anyone actually expected the model to solve it
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
AIs are gaining momentum, and "human level" is an inexistent milestone.
AI has now solved a major open problem -- one of the best known Erdos problems called the unit distance problem, one of Erdos's favourite questions and one that many mathematicians had tried. https://openai.com/index/model-disproves-discrete-geometry-conjecture/
So it took 20 months to go from making these plots on AIME problems to making them on 80 year old conjectures in combinatorial geometry…

A general purpose model made this breakthrough at the heart of geometry.
Exciting time ahead and probably no need for specialized models here!
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
A monumental achievement for AI. The wall falls first brick by brick, and then all of a sudden
AI has now solved a major open problem -- one of the best known Erdos problems called the unit distance problem, one of Erdos's favourite questions and one that many mathematicians had tried. https://openai.com/index/model-disproves-discrete-geometry-conjecture/
1/ Ten months ago, I was ecstatic that AI could win IMO gold.
Today, that excitement feels quaint: an internal @OpenAI model has refuted Erdos’s unit distance conjecture—a research result that one could recommend “acceptance without any hesitation” to the Annals of Mathematics.
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
2/ Why does this matter?
First, that we are here less than a year after IMO gold is a surprise to me. As bullish as I’ve been on AI math, I thought it would have taken longer to go from the 1.5 hour horizon of IMO proofs to the hundreds of hours needed for breakthrough research.

1/ Ten months ago, I was ecstatic that AI could win IMO gold. Today, that excitement feels quaint: an internal @OpenAI model has refuted Erdos’s unit distance conjecture—a research result that one could recommend “acceptance without any hesitation” to the Annals of Mathematics.
3/ In hindsight, it's not crazy that AI can shortcut these time horizons significantly: LLMs have superhuman knowledge bases and are primed to make insights that span research communities e.g. applying modern class field theory to discrete geometry in our case. Progress is fast!
2/ Why does this matter? First, that we are here less than a year after IMO gold is a surprise to me. As bullish as I’ve been on AI math, I thought it would have taken longer to go from the 1.5 hour horizon of IMO proofs to the hundreds of hours needed for breakthrough research.
5/5 This of course hits close to home: I’ve certainly seen my own research workflow transform over the past ~6 months.
For further commentary and contextualization on the math, check out the companion paper by the experts: https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-remarks.pdf
4/ Second, math is a leading indicator of what is to come. Soon—perhaps sooner than we all think—AI will begin autonomously producing landmark results in CS, physics, econ, bio, … We should be prepared for a new world where the nature and methods of science will have changed.
4/ Second, math is a leading indicator of what is to come. Soon—perhaps sooner than we all think—AI will begin autonomously producing landmark results in CS, physics, econ, bio, … We should be prepared for a new world where the nature and methods of science will have changed.
3/ In hindsight, it's not crazy that AI can shortcut these time horizons significantly: LLMs have superhuman knowledge bases and are primed to make insights that span research communities e.g. applying modern class field theory to discrete geometry in our case. Progress is fast!
Smile: a renaissance is upon us.
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
Frog should apologize to caterpillars
AI has now solved a major open problem -- one of the best known Erdos problems called the unit distance problem, one of Erdos's favourite questions and one that many mathematicians had tried. https://openai.com/index/model-disproves-discrete-geometry-conjecture/
From the post. 'The proof came from a new general-purpose reasoning model' 'An internal OpenAI model' And what might the name of this model be?

Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
Proof PDF: https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf
From the post. 'The proof came from a new general-purpose reasoning model' 'An internal OpenAI model' And what might the name of this model be?
'This result marks an important moment in the interaction between AI and mathematics: an AI system has autonomously resolved a longstanding open problem at the center of an active field. It also offers an early glimpse of a new kind of collaboration between AI and human mathematicians. In this case, the companion work by external mathematicians paints a substantially richer picture than the original solution alone.'

Proof PDF: https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf
Today, we’re sharing that a general-purpose internal @openai model achieved a breakthrough on one of the best-known combinatorial geometry problems. Less than 1 year ago frontier AI models were at IMO gold-level performance. I expect this pace of progress to continue.
'This is a general-purpose LLM. It wasn't targeted at this problem or even at mathematics. Also, it's not a scaffold.'
Emergent, like Mythos.
This is a general-purpose LLM. It wasn’t targeted at this problem or even at mathematics. Also, it’s not a scaffold. We have not pushed this model to the limit on open problems. Our focus is to get it out quickly so that everyone can use it for themselves.
An internal OpenAI model has disproved one of the most well-known Erdős problems: the unit distance problem. This is, without doubt, the most impressive achievement of AI in mathematics so far. https://openai.com/index/model-disproves-discrete-geometry-conjecture/
'math is a leading indicator of what is to come. Soon-perhaps sooner than we all think-Al will begin autonomously producing landmark results in CS, physics, econ, bio, ... We should be prepared for a new world where the nature and methods of science will have changed.'
4/ Second, math is a leading indicator of what is to come. Soon—perhaps sooner than we all think—AI will begin autonomously producing landmark results in CS, physics, econ, bio, … We should be prepared for a new world where the nature and methods of science will have changed.
If you are a mathematician, then you may want to make sure you are sitting down before reading further.
http://x.com/i/article/2057150538202976256
Narrators voice 'The name of this model? GPT-5.6'
@deanwball 'Rejoice, my friends, or weep with sorrow. What California is today, the world will be tomorrow.'
Smile: a renaissance is upon us.
@polynoamial @OpenAI Congrats!
Today, we’re sharing that a general-purpose internal @openai model achieved a breakthrough on one of the best-known combinatorial geometry problems. Less than 1 year ago frontier AI models were at IMO gold-level performance. I expect this pace of progress to continue.
@_aidan_clark_ Great time to be alive. Congrats!
Let's break this down, step by step [...]
waiting with bated breath for Gary’s take
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
OpenAI for research!
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
Yeah, this is now getting real.
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
@ChrSzegedy Yup, I think you nailed that prediction. What are some of your other predictions?
Whatever the definition of "superhuman AI mathematician" is, I think my original prediction of June 2026 is not too far off the mark.
Where were you when AI disproved the Erdős planar conjecture?
@_aidan_clark_ > After verifying the initial proof, we investigated the success rate of our models on this problem with varying amounts of test-time compute. The results are shown here.
this is the verification accuracy here right? or derivation of the proof again?
Let's break this down, step by step [...]
ask your codex what is extraordinary about this proof to feel it
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
wtf wtf wtf
🧵(1/8) An @OpenAI internal reasoning LLM achieved an AI Math milestone: solving an open problem central to its mathematical subfield— in this case, the unit distance problem of discrete geometry. We came across it in a side quest to truly push our model on the hardest problems.
Probably the best summary of OpenAI latest math breakthrough: "first example of a result produced autonomously by an AI that I find exciting in itself, as opposed to as a leading indicator." Feels like they’re really scaling search rather than solutions to bounded problems.
(What I wrote is screenshotted below.)
An internal general-purpose reasoning model at OpenAI just made a huge breakthrough.
Here is how Fields Medalist Timothy Gowers puts it: "What's significant about this moment is that it's the first really clear example of AI solving — not just an unsolved math problem — but a really well-known math problem."
the model was probably something like GPT-5.6-Pro-xhigh
An internal general-purpose reasoning model at OpenAI just made a huge breakthrough. Here is how Fields Medalist Timothy Gowers puts it: "What's significant about this moment is that it's the first really clear example of AI solving — not just an unsolved math problem — but a really well-known math problem."
@polynoamial not a scaffold => not a Pro model ?
This is a general-purpose LLM. It wasn’t targeted at this problem or even at mathematics. Also, it’s not a scaffold. We have not pushed this model to the limit on open problems. Our focus is to get it out quickly so that everyone can use it for themselves.
What a time
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
Congratulations to the team! The visualization is also so elegant
🧵(1/8) An @OpenAI internal reasoning LLM achieved an AI Math milestone: solving an open problem central to its mathematical subfield— in this case, the unit distance problem of discrete geometry. We came across it in a side quest to truly push our model on the hardest problems.
It’s a really special time to be alive…some thoughts from training this model 🧵
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
2/n Last year AI models achieved IMO gold level performance, but the jury was still out on whether they could do novel research. Today, our model has produced work that leading mathematicians like Tim Gowers said they would accept into Annals of Mathematics “without any hesitation.”
It’s a really special time to be alive…some thoughts from training this model 🧵
3/n Sometimes from the outside, it seems like we focus a lot on math. That's because math is a field where it is easy to share landmark results of this sort. However, the model that produced this is a general purpose model - it was not trained with the goal of doing math research.
2/n Last year AI models achieved IMO gold level performance, but the jury was still out on whether they could do novel research. Today, our model has produced work that leading mathematicians like Tim Gowers said they would accept into Annals of Mathematics “without any hesitation.”
6/n It only took 10 months to go from IMO gold to original math research. Working on this model and seeing what it can do every day has been very AGI-pilling for me, can’t wait to see where we are next year and time to lock tf in to make it happen!
I gave a talk with this slide less than two weeks ago and now I already have to update it. Crazy!!!

Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
another
(What I wrote is screenshotted below.)

Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
someone should probably turn this into a proper benchmark, same w mythos exploit cherrypicking

unfortunately openai didn't publish the unsummarized chain of thought, but the summary is 125 pages!
the model reaches the crucial idea (which it describes as 'frightening,' i would love to read the unabridged chain of thought here...) on page 39

Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
A new AI milestone today: "If a human had written the paper and submitted it to the Annals of Mathematics and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation. No previous AI-generated proof has come close to that.” - Tim Gowers 1/
the time is almost here.
Today, we share a breakthrough on the planar unit distance problem, a famous open question first posed by Paul Erdős in 1946. For nearly 80 years, mathematicians believed the best possible solutions looked roughly like square grids. An OpenAI model has now disproved that belief, discovering an entirely new family of constructions that performs better. This marks the first time AI has autonomously solved a prominent open problem central to a field of mathematics.
@polynoamial please do.
This is a general-purpose LLM. It wasn’t targeted at this problem or even at mathematics. Also, it’s not a scaffold. We have not pushed this model to the limit on open problems. Our focus is to get it out quickly so that everyone can use it for themselves.


