An OpenAI internal general-purpose reasoning model refutes Erdős’s 1946 unit-distance conjecture by identifying infinite families of point configurations with superlinear scaling
The proof is strong enough for submission to the Annals of Mathematics.
#8Sam Altman@SAMAa general-purpose model solved a major open problem in mathematics.
we'll be saying this a lot over the coming years, but this is a kinda big milestone.
i'm very excited for AI to greatly extend our understanding of the world, but still, i have complicated feelings today.
Timothy Gowers @wtgowers@wtgowersIf you are a mathematician, then you may want to make sure you are sitting down before reading further.
#19Greg Brockman@GDBAn 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.
OpenAI@OpenAIToday, 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.
#19Greg Brockman@GDBour math result is a milestone in new knowledge generation by AI. very exciting to imagine similar results in other scientific fields. "It's very hard to sleep, man" is a pretty good reaction.
Alex Dimakis@AlexGDimakisA breakthrough by OpenAI in a very famous Combinatorics problem, the Planar Unit Distance problem by Erdos 1946. The problem is amazing because it can be described to a first-grader: Find a way to place n points on the plane to maximize the number of pairs that have distance exactly 1. For example, if you have n=4 points on a square (of side-length 1) you have 4 pairs of distance 1. The diagonals have length sqrt(2) so don't count. But you can squeeze one diagonal and create a point-set with n=4 points and 5 pairs of distance 1. And you can't get more than 5 pairs from n=4 points, so we are done with n=4 points. Now, if you place n points on a line, you have n-1 pairs of distance 1. In general, all known constructions of n points had a number of pairs scaling essentially linearly: n^{1+something vanishing} It seems that the model found a way to place n points on the plane so that their unit distances scale super-linearly: like n^{1+delta} for some *constant* delta. Delta was not explicitly specified apparently, but a forthcoming refinement by Will Sawin shows delta=0.014 works, according to the announcement. This is incredible progress for mathematics, since this is (unlike previous Erdos problems solved by AI) a major breakthrough, in one of the most studied problems in combinatorial geometry. If you're in mathematics research now, you feel the AGI. Lijie Chen said it honestly in the video: "It's very hard to sleep, man"
#31Noam Brown@POLYNOAMIALToday, 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.
OpenAI@OpenAIToday, 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.
Noam Brown@polynoamialToday, 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.
Since people are asking, no it did not use Lean. But I don't think it should matter anyway.
Noam Brown@polynoamialThis 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.
#31Noam Brown@POLYNOAMIALToday, 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.
OpenAI@OpenAIToday, 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.
#31Noam Brown@POLYNOAMIALExcellent thread from mathematician Tim Gowers on the significance of the @OpenAI model’s breakthrough on the Erdos Unit Distance Problem!
Timothy Gowers @wtgowers@wtgowersIf you are a mathematician, then you may want to make sure you are sitting down before reading further.
#36Eric Jang@ERICJANG11[x] automated math machine [ ] proof / disproof of navier stokes conjecture [ ] recursive self improver
Prediction: all this and more will be accomplished by EOY 2026
Alexander Wei@alexwei_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.
#36Eric Jang@ERICJANG11amazing
Sebastien Bubeck@SebastienBubeckhttp://x.com/i/article/2057150538202976256
@BorisMPower Congrats to the result, still disagreeing with the sentiment.
Boris Power@BorisMPowerA general purpose model made this breakthrough at the heart of geometry. Exciting time ahead and probably no need for specialized models here!
#43Christian Szegedy@CHRSZEGEDYWhatever the definition of "superhuman AI mathematician" is, I think my original prediction of June 2026 is not too far off the mark.
Timothy Gowers @wtgowers@wtgowersAI 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.
Bojan Tunguz@tunguz@ChrSzegedy Yup, I think you nailed that prediction. What are some of your other predictions?
@kareem_carr There was 0 human involvement. The prompt is in the report. The final answer by the model is in the report. And we have a (gpt-rewritten) CoT that we released.
#55Lucas Beyer (bl16)@GIFFMANAAh, there it is! I was already getting worried that they didn't have any IO announcement this year
OpenAI@OpenAIToday, 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.
#56Dan Roy@ROYDANROYCongrats to OpenAI on their breakthrough in discrete geometry.
Timothy Gowers @wtgowers@wtgowersAI 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/
#72Ofir Press@OFIRPRESSNoga Alon's comment about the new result:
OpenAI@OpenAIToday, 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.
#92(((ل()(ل() 'yoav))))👾@YOAVGOwhat i infer from this is that the ai-mathematician curses a lot and is somewhat toxic
Sebastien Bubeck@SebastienBubeck@kareem_carr There was 0 human involvement. The prompt is in the report. The final answer by the model is in the report. And we have a (gpt-rewritten) CoT that we released.
@SebastienBubeck @kareem_carr can you release also the rewriting prompt?
Sebastien Bubeck@SebastienBubeck@kareem_carr There was 0 human involvement. The prompt is in the report. The final answer by the model is in the report. And we have a (gpt-rewritten) CoT that we released.
@SuryaGanguli parroting stochastic parrots on bluesky
Surya Ganguli@SuryaGanguliWhere are the stochastic parrot folks at? 😉
#153Gary Marcus@GARYMARCUSnor do we know how the (new) model works nor how it does on anything else nor how it was trained.
scientists wait for facts; cheerleaders (over and over) rush to judgments that have often been wrong.
let’s see what we actually have here.
@polynoamial did it use tools like Lean?
Noam Brown@polynoamialThis 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.
@polynoamial not even to generate augmented data?
Noam Brown@polynoamialSince people are asking, no it did not use Lean. But I don't think it should matter anyway.
@polynoamial also are you saying that the only thing is novel is scale?
Noam Brown@polynoamialSince people are asking, no it did not use Lean. But I don't think it should matter anyway.
#154Andrew Gordon Wilson@ANDREWGWILSIt's an exciting time to be alive. In the space of a couple decades, I think AI has the potential to accelerate scientific progress by hundreds of years. I've always wanted to time travel, just so I can ask the big questions. Maybe I won't have to?
Timothy Gowers @wtgowers@wtgowersAI 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/
#167Emad@EMOSTAQUEOnce 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?
OpenAI@OpenAIToday, 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 breakthrough by OpenAI in a very famous Combinatorics problem, the Planar Unit Distance problem by Erdos 1946.
The problem is amazing because it can be described to a first-grader: Find a way to place n points on the plane to maximize the number of pairs that have distance exactly 1.
For example, if you have n=4 points on a square (of side-length 1) you have 4 pairs of distance 1. The diagonals have length sqrt(2) so don't count. But you can squeeze one diagonal and create a point-set with n=4 points and 5 pairs of distance 1. And you can't get more than 5 pairs from n=4 points, so we are done with n=4 points.
Now, if you place n points on a line, you have n-1 pairs of distance 1. In general, all known constructions of n points had a number of pairs scaling essentially linearly: n^{1+something vanishing}
It seems that the model found a way to place n points on the plane so that their unit distances scale super-linearly: like n^{1+delta} for some *constant* delta. Delta was not explicitly specified apparently, but a forthcoming refinement by Will Sawin shows delta=0.014 works, according to the announcement.
This is incredible progress for mathematics, since this is (unlike previous Erdos problems solved by AI) a major breakthrough, in one of the most studied problems in combinatorial geometry. If you're in mathematics research now, you feel the AGI. Lijie Chen said it honestly in the video: "It's very hard to sleep, man"
@SebastienBubeck @kareem_carr Was this a single shot, or an agent with python etc that tried things and wrote code?
Sebastien Bubeck@SebastienBubeck@kareem_carr There was 0 human involvement. The prompt is in the report. The final answer by the model is in the report. And we have a (gpt-rewritten) CoT that we released.
@SebastienBubeck Great explanation, thanks for posting and Congratulations.
Sebastien Bubeck@SebastienBubeckhttp://x.com/i/article/2057150538202976256
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/
Ethan Mollick@emollickJune 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"
#175Ethan Mollick@EMOLLICKThe frontier is still jagged though (here is Gemini 3.5 Flash messing up counting letters in words)
#175Ethan Mollick@EMOLLICKIf this is true, using the best public estimates we have of LLM resource use, solving this Erdos problem took 0.6–6.3 kWh of electricity and about 3–31 liters of water.
So that is less than three almonds worth of water and the electricity equivalent of 2-20 miles of EV driving.
will depue@willdepuejust quick napkin math on how long this took (unless i missed where they said): the published CoT summary is 111,145 tokens long. it's really hard to say how much they summarized, assume 3x-20x reduction in tokens? and i'm assuming this is gpt-5.6 pro, so taking Artifical Analysis' benchmark of 51ms tok/sec at 100k input for gpt 5.5. underestimate prob hard to say this seems a bit low so going to multiply all of this by 2x then this probably took anywhere between 5 hours to 32 hours. so like $120 - $1000 in gpt 5.5 pro tokens whole point is not that long for a result of this magnitude!
Estimates of power usage here: https://arxiv.org/pdf/2509.20241 (these numbers also match independent assessments)
Estimates of water usage here: https://eta-publications.lbl.gov/sites/default/files/2024-12/lbnl-2024-united-states-data-center-energy-usage-report_1.pdf (note it only includes direct cooling, not water for electricity generation)
Ethan Mollick@emollickIf this is true, using the best public estimates we have of LLM resource use, solving this Erdos problem took 0.6–6.3 kWh of electricity and about 3–31 liters of water. So that is less than three almonds worth of water and the electricity equivalent of 2-20 miles of EV driving.
Individual use is small, but at aggregate scale, resource usage is higher. By 2030, AI may use as much electricity as Japan.
Water use will remain less than 1% of total US water use in 2030, but that can still strain local utilities.
(and this problem alone took many runs)
Ethan Mollick@emollickEstimates of power usage here: https://arxiv.org/pdf/2509.20241 (these numbers also match independent assessments) Estimates of water usage here: https://eta-publications.lbl.gov/sites/default/files/2024-12/lbnl-2024-united-states-data-center-energy-usage-report_1.pdf (note it only includes direct cooling, not water for electricity generation)
#175Ethan Mollick@EMOLLICKIts The Graph again (not the METR graph, the one from the o1 launch).
Although no logarithmic decay of ability with increasing compute...
Noam Brown@polynoamialThis 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.
#187Michael Nielsen@MICHAEL_NIELSENVery striking, as is the linked post:
Timothy Gowers @wtgowers@wtgowersAI 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/
#241Aidan Clark@_AIDAN_CLARK_Let's break this down, step by step [...]
OpenAI@OpenAIToday, 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.
@eliebakouch y-axis is “percentage of the time you successfully derive the counterexample/proof”, not just verification.
elie@eliebakouch@_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?
#254will depue@WILLDEPUEit’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
will depue@willdepuewhat a moment. wow. a bit in shock
#254will depue@WILLDEPUEwhat a moment. wow. a bit in shock
OpenAI@OpenAIToday, 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.
#254will depue@WILLDEPUEi’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
OpenAI@OpenAIToday, 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.
#254will depue@WILLDEPUEjust quick napkin math on how long this took (unless i missed where they said): the published CoT summary is 111,145 tokens long. it's really hard to say how much they summarized, assume 3x-20x reduction in tokens and i'm assuming this is gpt-5.6 pro, so taking Artifical Analysis' benchmark of 51ms tok/sec at 100k input for gpt 5.5 then this probably took anywhere between 2.5 hours to 16 hours. not that long for a result of this magnitude! so like $60 - $500 in gpt 5.5 pro tokens
OpenAI@OpenAIToday, 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.
#254will depue@WILLDEPUEjust quick napkin math on how long this took (unless i missed where they said): the published CoT summary is 111,145 tokens long. it's really hard to say how much they summarized, assume 3x-20x reduction in tokens? and i'm assuming this is gpt-5.6 pro, so taking Artifical Analysis' benchmark of 51ms tok/sec at 100k input for gpt 5.5. underestimate prob hard to say this seems a bit low so going to multiply all of this by 2x then this probably took anywhere between 5 hours to 32 hours. so like $120 - $1000 in gpt 5.5 pro tokens whole point is not that long for a result of this magnitude!
OpenAI@OpenAIToday, 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.
@AndrewCurran_ yeah just assuming from 'general purpose model' and 'we're going to make this accessible as soon as possible' it sounds like just the next iteration of the model in pro mode
Andrew Curran@AndrewCurran_'im assuming this is GPT-5.6 Pro'
#256Surya Ganguli@SURYAGANGULIWhere are the stochastic parrot folks at? 😉
Timothy Gowers @wtgowers@wtgowersAI 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/
#263Andrew Carr 🤸@ANDREW_N_CARR`This may indicate one way that AI systems have an edge: it’s not just that they can try all known methods, but they can play for longer and in more treacherous waters than mathematicians without getting overwhelmed`
OpenAI@OpenAIToday, 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.
#331François Fleuret@FRANCOISFLEURETAIs are gaining momentum, and "human level" is an inexistent milestone.
Timothy Gowers @wtgowers@wtgowersAI 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/
@willdepue It’s 90 in the Erdos list, they probably tried as part of trying everything?
will depue@willdepuei’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
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…
#357Boris Power@BORISMPOWERA general purpose model made this breakthrough at the heart of geometry.
Exciting time ahead and probably no need for specialized models here!
OpenAI@OpenAIToday, 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.
#364Mo Bavarian@MOBAV0A monumental achievement for AI. The wall falls first brick by brick, and then all of a sudden
Timothy Gowers @wtgowers@wtgowersAI 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/
#367Alexander Wei@ALEXWEI_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.
OpenAI@OpenAIToday, 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.
Alexander Wei@alexwei_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!
Alexander Wei@alexwei_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
Alexander Wei@alexwei_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.
Alexander Wei@alexwei_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!
#392Dean W. Ball@DEANWBALLSmile: a renaissance is upon us.
OpenAI@OpenAIToday, 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.
#406Yu Bai@YUBAI01Apart from the significance of the result, what makes this encouraging is that the model training was not specifically optimized for math research -- it is a generally capable model and this result is one magic we get out of it.
OpenAI@OpenAIToday, 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
Timothy Gowers @wtgowers@wtgowersAI 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/
a word of yap yap cock-a-doodle-doo How much does it cost to solve this kind of problem manually? Can you put a number on the value of a solution that best human efforts have failed to attain? Can you put a number on the value of a *general capability* to attain such solutions?
#517Andrew Curran@ANDREWCURRAN_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?
OpenAI@OpenAIToday, 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
Andrew Curran@AndrewCurran_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.'
Andrew Curran@AndrewCurran_Proof PDF: https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf
#517Andrew Curran@ANDREWCURRAN_Noam Brown@polynoamialToday, 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.
#517Andrew Curran@ANDREWCURRAN_'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.
Noam Brown@polynoamialThis 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.
#517Andrew Curran@ANDREWCURRAN_
Thomas Bloom@thomasfbloomAn 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/
#517Andrew Curran@ANDREWCURRAN_'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.'
Alexander Wei@alexwei_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.
#517Andrew Curran@ANDREWCURRAN_Timothy Gowers @wtgowers@wtgowersIf you are a mathematician, then you may want to make sure you are sitting down before reading further.
#517Andrew Curran@ANDREWCURRAN_Sebastien Bubeck@SebastienBubeckhttp://x.com/i/article/2057150538202976256
Narrators voice 'The name of this model? GPT-5.6'
Andrew Curran@AndrewCurran_#517Andrew Curran@ANDREWCURRAN_'im assuming this is GPT-5.6 Pro'
will depue@willdepuejust quick napkin math on how long this took (unless i missed where they said): the published CoT summary is 111,145 tokens long. it's really hard to say how much they summarized, assume 3x-20x reduction in tokens? and i'm assuming this is gpt-5.6 pro, so taking Artifical Analysis' benchmark of 51ms tok/sec at 100k input for gpt 5.5. underestimate prob hard to say this seems a bit low so going to multiply all of this by 2x then this probably took anywhere between 5 hours to 32 hours. so like $120 - $1000 in gpt 5.5 pro tokens whole point is not that long for a result of this magnitude!
@willdepue I agree!
will depue@willdepue@AndrewCurran_ yeah just assuming from 'general purpose model' and 'we're going to make this accessible as soon as possible' it sounds like just the next iteration of the model in pro mode
@deanwball 'Rejoice, my friends, or weep with sorrow. What California is today, the world will be tomorrow.'
Dean W. Ball@deanwballSmile: a renaissance is upon us.
@voooooogel @zacharynado The wonderful terror of realizing its own strength.
thebes@voooooogelunfortunately 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
#548Clive Chan@ITSCLIVETIME🚀
Timothy Gowers @wtgowers@wtgowersAI 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/
@polynoamial @OpenAI Congrats!
Noam Brown@polynoamialToday, 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!
Aidan Clark@_aidan_clark_Let's break this down, step by step [...]
#597Patrick Hsu@PDHSUamazing time to be alive
OpenAI@OpenAIToday, 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.
#604Simo Ryu@CLONEOFSIMOJust three years ago some people were certain these models will not have genuine, out of distribution capability. What an incredible achievement. It almost feels like end game.
OpenAI@OpenAIToday, 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.
#604Simo Ryu@CLONEOFSIMOJust three years ago some people were certain these models will not have genuine, out of distribution capability. What an incredibly achievement.
OpenAI@OpenAIToday, 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.
#638Yo Shavit@YONASHAVwaiting with bated breath for Gary’s take
OpenAI@OpenAIToday, 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.
@emollick @alexolegimas And they still can’t count without problem-specific hacks.
Ethan Mollick@emollickJune 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"
#672Jason Phang@ZHANSHENGOpenAI for research!
OpenAI@OpenAIToday, 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.
#677Jerry Liu@JERRYJLIU0proof too complicated, Claude help ELI5
OpenAI@OpenAIToday, 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.
#687Bojan Tunguz@TUNGUZYeah, this is now getting real.
OpenAI@OpenAIToday, 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?
Christian Szegedy@ChrSzegedyWhatever 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?
#716elie@ELIEBAKOUCH@_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?
Aidan Clark@_aidan_clark_Let's break this down, step by step [...]
#716elie@ELIEBAKOUCH@_aidan_clark_ ok so it's basically the same setup as when the model derive it for the first time right? really cool plot
Aidan Clark@_aidan_clark_@eliebakouch y-axis is “percentage of the time you successfully derive the counterexample/proof”, not just verification.
#732Chris Paxton@CHRIS_J_PAXTONI guess this is what living through the singularity would look like huh
Ethan Mollick@emollickJune 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"
#757Ekin Akyürek@AKYUREKEKINask your codex what is extraordinary about this proof to feel it
OpenAI@OpenAIToday, 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.
#801Sherwin Wu@SHERWINWUThis result is cool partially because of how directly it ties to the OpenAI mission ("ensuring that AGI benefits all of humanity").
Solving open math problems – which literally advances _all_ of humanity forward – is one of the purest applications of that mission. Wild!
OpenAI@OpenAIToday, 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.
#812Brendan Dolan-Gavitt@MOYIXAll AI can do is plagiarize, here we see it regurgitating one of the proofs from The Book
EigenGender 🔸@EigenGendernot impressive, the conjecture was already in the training data
#826You Jiacheng@YOUJIACHENGwtf wtf wtf
Hongxun Wu@HongxunWu🧵(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.
#826You Jiacheng@YOUJIACHENGthe cost is surprisingly low
will depue@willdepuejust quick napkin math on how long this took (unless i missed where they said): the published CoT summary is 111,145 tokens long. it's really hard to say how much they summarized, assume 3x-20x reduction in tokens? and i'm assuming this is gpt-5.6 pro, so taking Artifical Analysis' benchmark of 51ms tok/sec at 100k input for gpt 5.5. underestimate prob hard to say this seems a bit low so going to multiply all of this by 2x then this probably took anywhere between 5 hours to 32 hours. so like $120 - $1000 in gpt 5.5 pro tokens whole point is not that long for a result of this magnitude!
#836kalomaze@KALOMAZEit is funny how weakly calibrated frontier models are on how fast their own progression is moving
OpenAI@OpenAIToday, 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.
#837Jimmy Apples 🍎/acc@APPLES_JIMMYAnd with a general purpose model too.
Remember when you were playing around with 3.5 and now a few years later we have this.
What’s the next math breakthrough ?
Millennium problems still seem a year or more out.
OpenAI@OpenAIToday, 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.
#851Zhuohan Li@ZHUOHAN123Exciting time to be alive!
OpenAI@OpenAIToday, 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.
#867Alexander Doria@DORIALEXANDERProbably 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.
Daniel Litt@littmath(What I wrote is screenshotted below.)
.@voooooogel singled out the specific passage where the model gets on track to the final solution and, yeah, definitely conveyed the thrill, emotion and vertigo of something new.
Alexander Doria@DorialexanderProbably 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.
@VictorTaelin Could be RLM-style but maybe more prosaically: they have the longer context they could not deploy commercially yet.
Taelin@VictorTaelinthis is super cool but I still do not understand how they get a model to coherently and usefully reason for that amount tokens and at this point I'm to afraid to ask
#909Chi Jin@CHIJINMLVery proud to have contributed to the training of this OpenAI internal model, which achieved this mathematical breakthrough! What’s surprising and amazing is that it’s truly a general-purpose model: not specially trained for math, and using no scaffolding.
Noam Brown@polynoamialToday, 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.
#909Chi Jin@CHIJINML@tenobrus depends on what you viewed as scaffolding, definitely no specialized scaffolding for math.
Tenobrus@tenobrus@chijinML hi chi, can you help clarify: when you say "no scaffolding" does that mean no very mathematically specialized scaffolding? no tool calls at all? did the model use Lean in any way? or was it genuinely just one really massive chain of thought rollout?
#927Daniel@GROWING_DANIELKind of disturbing honestly. There’s something about God in math proofs so this is a weird moment.
Having an AI model solve a famous problem feels way more monumental to me than anything else so far.
OpenAI@OpenAIToday, 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.
These stochastic parrot got hands
OpenAI@OpenAIToday, 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.
#980Lisan al Gaib@SCALING01An 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
Lisan al Gaib@scaling01An 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 ?
Noam Brown@polynoamialThis 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.
#1032Rohan Paul@ROHANPAUL_AIAI in math is creating history again, as OpenAI's general-purpose reasoning model has disproved a major Erdős conjecture from 1946.
The important part is not that AI solved a hard math problem, but how little special machinery it needed.
For decades, the planar unit distance problem looked almost embarrassingly simple: place points on a plane, then ask how many pairs can be exactly one unit apart.
For decades, the best examples looked like stretched versions of a square grid, so mathematicians believed grids were almost the best possible design.
OpenAI’s internal model broke that picture by finding an infinite family of constructions that gives a polynomial improvement, with the proof checked by external mathematicians.
The point to note is that the model was not a bespoke theorem-proving engine trained only for this problem, and the official post says its success improved with more test-time compute, meaning more reasoning at inference rather than only more training.
That matters so much, because research progress often comes from holding a fragile chain of ideas together long enough to cross from one field into another.
In this case, the bridge ran from a plain geometric question into deep algebraic number theory, including machinery like infinite class field towers and Golod–Shafarevich theory.
And now we see a general-purpose reasoning system appears able to search a conceptual space where human taste, field boundaries, and inherited guesses may have quietly narrowed the path.
So future is not machines replacing judgment, but machines widening the map before judgment begins.
OpenAI@OpenAIToday, 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.
#1047Taelin@VICTORTAELINthis is super cool but I still do not understand how they get a model to coherently and usefully reason for that amount tokens and at this point I'm to afraid to ask
thebes@voooooogelunfortunately 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
#1054Anders Sandberg@ANDERSSANDBERGThis is impressive: it is a problem I had actually heard of. It looks like the solution approach is surprising to mathematicians. It was a general reasoning model rather than a specialized one: bitter lesson time. I think the stochastic parrot is now nuked from orbit.
Timothy Gowers @wtgowers@wtgowersAI 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/
One can quibble. The initial proof was improved by humans to something tighter. It is still not a problem or real-world importance. Math might be particularly amenable to AI. Or combinatorics. But still...
Anders Sandberg@anderssandbergThis is impressive: it is a problem I had actually heard of. It looks like the solution approach is surprising to mathematicians. It was a general reasoning model rather than a specialized one: bitter lesson time. I think the stochastic parrot is now nuked from orbit.
The above plot is interesting. Right now centaurs rule. I wonder how long before the blue curve starts overtaking it? Also, Claude noted that the April 9 burst is an apparent batched release from OpenAI "internal model", perhaps the same one as this.
Anders Sandberg@anderssandbergLast year we were impressed that AI could find forgotten proofs of conjectures in literature. Then solve minor Erdös conjectures. Then actually doing it with interesting new approaches. Now solving a conjecture people have heard of.
Last year we were impressed that AI could find forgotten proofs of conjectures in literature. Then solve minor Erdös conjectures. Then actually doing it with interesting new approaches. Now solving a conjecture people have heard of.
Anders Sandberg@anderssandbergOne can quibble. The initial proof was improved by humans to something tighter. It is still not a problem or real-world importance. Math might be particularly amenable to AI. Or combinatorics. But still...
#1088Andrea Montanari@ANDREA__MI can recognize some faces here 😀
OpenAI@OpenAIToday, 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.
#1135Cheng Lu@CLU_CHENGWhat a time
OpenAI@OpenAIToday, 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.
#1220rohit@KRISHNANROHITI'd love for a model to prove a graph theory problem to take existing models and what's in their training to identify what's the most likely "adjacent possible" problem that will get cracked next.
Timothy Gowers @wtgowers@wtgowersAI 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/
#1240Boyuan Chen@BOYUANCHEN0Congratulations to the team! The visualization is also so elegant
Hongxun Wu@HongxunWu🧵(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.
#1245Alex Volkov@ALTRYNEWe're about to find out if we live in a simulation, very soon.
@demishassabis just said "we're at the foothills of the singularity" and now OpenAI announces first novel Math problems solutions by AI!
Very much looking forward to breakthroughs in physics next!
OpenAI@OpenAIToday, 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.
#1270Sheryl Hsu@SHERYLHSU02It’s a really special time to be alive…some thoughts from training this model 🧵
OpenAI@OpenAIToday, 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.”
Sheryl Hsu@SherylHsu02It’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.
Sheryl Hsu@SherylHsu022/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.”
#1270Sheryl Hsu@SHERYLHSU026/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!
#1271Damek@DAMEKDAVISI gave a talk with this slide less than two weeks ago and now I already have to update it. Crazy!!!
OpenAI@OpenAIToday, 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.
#1271Damek@DAMEKDAVISPlease try all of these with the internal model as well:
#1271Damek@DAMEKDAVISanother
Daniel Litt@littmath(What I wrote is screenshotted below.)
#1332Prakash@8TEAPIroughly a month late… and directly from OpenAI using a general model rather than a scaffold company
OpenAI@OpenAIToday, 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 @OpenAI This is impressive
Noam Brown@polynoamialToday, 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.
#1333Shaun Maguire@SHAUNMMAGUIREMost predictions I've made about hardware timelines have been too optimistic
Most predictions I've made about AI timelines have been too pessimistic
This field is moving so quickly
Incredible milestone
OpenAI@OpenAIToday, 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.
#1402bilal@BILALTWOVECwhat did 5.6 pro see
thebes@voooooogelunfortunately 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
#1402bilal@BILALTWOVEC
OpenAI@OpenAIToday, 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
bilal@bilaltwovec
#1446Shubhendu Trivedi@_ONIONESQUEWow this is exciting! This is a famous problem from a beautiful area (Szemeredi-Trotter, Crossing Lemma, Polynomial Ham Sandwich are all in the vicinity), and yet the construction of the family of counterexamples comes from an unexpected connection from algebraic number theory.
Timothy Gowers @wtgowers@wtgowersAI 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/
The proof is something I am in no position to begin to understand, of course, and the note posted mentions that the direction is not altogether new. However, what I do know is that operators in the above broader area would be unlikely to make the connection.
Shubhendu Trivedi@_onionesqueWow this is exciting! This is a famous problem from a beautiful area (Szemeredi-Trotter, Crossing Lemma, Polynomial Ham Sandwich are all in the vicinity), and yet the construction of the family of counterexamples comes from an unexpected connection from algebraic number theory.
@roydanroy Combinatorialists and incidence / discrete geometry experts wouldn't have any Algebraic number theory chops. AI models can pattern match wherever they want. Erdos also believing that the conjecture was true biased folk. The LM could run amok in either direction.
Dan Roy@roydanroyCongrats to OpenAI on their breakthrough in discrete geometry.
#1460Jiaxin Wen@JIAXINWEN22time will tell
Jiaxin Wen@jiaxinwen22I might be one of the few people who is most bearish on human research taste and bullish on automated research: - "AIs can only do hyperparameter search" is mainly a skill issue with bad automated research setups. - human taste is overrated, e.g. frontier labs / neolabs are doing pretty simlar things. - human taste might win in a low-compute world, but not a high-compute world we're entering.
#1488Samuel Hammond 🦉@HAMANDCHEESEMysterium Tremendum
thebes@voooooogelunfortunately 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
#1496Chubby♨️@KIMMONISMUSOpenAI made history today.
An internal reasoning model autonomously disproved a famous conjecture in mathematics that stood for nearly 80 years.
The problem: In 1946, Paul Erdős asked how many pairs of points can be exactly 1 unit apart if you place n points on a flat surface. The best known answer came from square grid constructions, and Erdős himself conjectured you can't do meaningfully better. Mathematicians believed this for decades.
The AI proved him wrong. It found entirely new point configurations that beat the square grid by a fixed polynomial factor, not a marginal improvement, a real mathematical gap.
The proof uses methods from algebraic number theory, a completely different branch of math, Class field towers, Golod-Shafarevich theory, tools nobody expected to be relevant to a geometry problem about distances in the plane (reminds me of move 37, AlphaGo tbh).
Fields Medalist Tim Gowers calls it "a milestone in AI mathematics." The proof was verified by leading external mathematicians.
According to OpenAI, this is the first time AI has independently solved a prominent open research problem in mathematics!
Caveat: Obviously OpenAI chose which problems to test the model on. So "autonomous" means the model generated the idea and wrote the proof, not that it wandered into the problem on its own.
But if reasoning models can reliably make cross-domain connections like this, finding paths that experts didn't prioritize, this changes research far beyond math. Biology, physics, materials science, medicine.
This isn't AI reproducing human knowledge anymore. This is AI producing new knowledge. That's a qualitative shift.
OpenAI@OpenAIToday, 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.
#1496Chubby♨️@KIMMONISMUSOpenAI is aiming for a release of their upcoming general-purpose LLM.
„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 makes this so impressive is that a general-purpose LLM, not specifically trained for math or this problem, appears to get dramatically better simply by using more test-time compute!
OpenAI has a run.
Noam Brown@polynoamialThis 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.
#1499thebes@VOOOOOOGELexclusive internal footage of gpt solving the planar unit distance problem
thebes@voooooogelunfortunately 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
#1499thebes@VOOOOOOGELunfortunately 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
OpenAI@OpenAIToday, 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.
#1560Andrei Bursuc@ABURSUCAmazed by this but also by @ChrSzegedy's foresight who predicted such advances a while ago.
Timothy Gowers @wtgowers@wtgowersAI 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/
@polynoamial @OpenAI Amazing!! Congratulations to everyone
Noam Brown@polynoamialToday, 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.
@sama really amazing, big congrats to the teams
Sam Altman@samaa general-purpose model solved a major open problem in mathematics. we'll be saying this a lot over the coming years, but this is a kinda big milestone. i'm very excited for AI to greatly extend our understanding of the world, but still, i have complicated feelings today.
Timothy Nguyen@IAmTimNguyenA 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/
#1627Aviv Tamar@AVIVTAMAR1What a great time to be tenured.
Timothy Gowers @wtgowers@wtgowersAI 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/
#1711🍓🍓🍓@IRULETHEWORLDMOthe time is almost here.
OpenAI@OpenAIToday, 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.
Noam Brown@polynoamialThis 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.
#1884Ethan@TORCHCOMPILEDThis is posed as a “first ever” moment but I feel like I’m having Déjà vu? Are these other cases, this one in January, less impressive? Asking as a fella barely familiar with Erdos problems
OpenAI@OpenAIToday, 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.
#1891Lijie Chen@WJMZBMR11/ Today, an internal @OpenAI model has refuted Erdős’s unit distance conjecture — a research result that one could recommend “acceptance without any hesitation” to the Annals of Mathematics, one of the most prestigious journals in mathematics.
We came across it in a side quest to push our model on the hardest problems.
OpenAI@OpenAIToday, 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.
#1894Nick Dobos@NICKADOBOSChatGPT was frightened upon discovery of new math.
Curious behavior. I’m not sure I had an expectation for what an Ai should feel upon making a novel discovery, but fear is interesting.
Not surprise. Not shock. Not glee. Not curiosity.
Fear.
thebes@voooooogelunfortunately 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
