/Tech23d ago

OpenAI's internal reasoning model refutes the 1946 Erdős planar unit distance conjecture

Nicknamed Mythos, the model used algebraic field methods.

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Original post

Aaron Roth#643

Timothy Gowers @wtgowers@wtgowers

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/

Timothy Gowers @wtgowers@wtgowers

If you are a mathematician, then you may want to make sure you are sitting down before reading further.

12:04 PM · May 20, 2026 · 1.4M Views

Sentiment

Positive users celebrate frontier models like Claude Mythos and GPT-5.5 for delivering elegant solutions to the Erdős unit distance problem, while negative users dismiss the claims as overhyped marketing or outright lies.

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57.2%

Neg

42.8%

1,540 comments with sentiment.

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OpenAI@OpenAI

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.

23d10.2M23.3K7.7K

Sam Altman@sama

a 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@wtgowers

If you are a mathematician, then you may want to make sure you are sitting down before reading further.

23d623.1K6.2K572

Sebastien Bubeck@SebastienBubeck

http://x.com/i/article/2057150538202976256

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thebes@voooooogel

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

OpenAI@OpenAI

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.

23d460.5K2.4K802

Steve Newman@snewmanpv

I think we are in the process of discovering that humans are bad at mathematics.

A gibbon would scoff at an Olympic climber; the human body is not optimized for climbing. We're getting mounting evidence that our brain may be far from optimal for advanced math.

No disrespect to mathematicians. I was a two-time IMO silver medalist; I'm just smart enough to appreciate that some people are much, much smarter. But it's starting to look like math is somewhere on the midpoint of Moravec’s paradox; between chess (computers surpassed us some time back) and cooking (probably many years to go, for general capabilities). It's fairly hard for us, and so it looks like computers are going to surpass us.

AI math still has important weaknesses. For instance, AI systems have not yet shown any ability to identify interesting research directions, or develop new concepts on which further work can build. But they are starting to look superhuman in some respects. And once AI *starts* to become superhuman in some domain, we all know what happens next.

Timothy Gowers @wtgowers@wtgowers

23d580.2K2.2K621

Ethan Mollick@emollick

If 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@willdepue

just 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!

23d612.9K2.8K524

Greg Brockman@gdb

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.

OpenAI@OpenAI

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.

23d381.7K2.8K462

Noam Brown@polynoamial

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.

OpenAI@OpenAI

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.

23d385.1K2.3K350

QC@QiaochuYuan

at this point it is completely untenable to believe anything along the lines of “AI can only spit out an average of the training data.” that was already only a very rough way of understanding older models pre-reasoning, it was already obsoleted by o1 which released in 2024, and now it should be obviously and conclusively dead even if you haven’t been paying close attention. recursive self-improvement has barely even started and we are already here. even with the recent erdos problem solves you could argue that those were cherrypicked out of a large database for being neglected by humans. that cope is no longer available

now generalize the lesson: all other arguments that there is some essential human activity forever beyond the reach of AI are also cope, these are technical problems and the will and money and talent exists and is being deployed to solve them. artificial superintelligence is not a fairy tale. assume it’s coming and plan accordingly

Timothy Gowers @wtgowers@wtgowers

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Daniel@growing_daniel

Kind 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@OpenAI

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.

23d305.5K2K372

Maxwell Tabarrok@MTabarrok

the machine gods are discovering new sacred geometries and you're dooming?

Alvaro Lozano-Robledo@mathandcobb

Following up on the suggestion from Will Sawin, here is an illustration of the new configurations that disprove Erdos' unit distance conjecture (made with the help of ChatGPT 5.5 Thinking).

22d254K1.5K382

Anders Sandberg@anderssandberg

This 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@wtgowers

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Sholto Douglas@_sholtodouglas

Huge credit to the OAI team for solving the unit distance problem with 5.5 - it is now my go to example that models can in fact pull together disparate ideas into new discoveries.

As with all 4 minute miles, we had to try and cross it too! Turns out mythos solves it with a cute, simple proof. This implies some serious overhang in discoveries!

levent@__alpoge__

over the weekend i checked the obvious thing, which is whether mythos is able to solve the erdos unit distance problem, aka erdos problem #90. the answer is: yea

17d345.2K1.6K333

Ethan Mollick@emollick

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"

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Sheryl Hsu@SherylHsu02

It’s a really special time to be alive…some thoughts from training this model 🧵

OpenAI@OpenAI

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.

23d313.3K1.1K317

Bojan Tunguz@tunguz

Yeah, this is now getting real.

OpenAI@OpenAI

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.

23d426.8K1.1K347

Chubby♨️@kimmonismus

Erdős problem #90 has been open for decades. Over the weekend a mathematician tested whether Claude Mythos could solve it. It did.

But what caught my attention: Mythos didn't replicate the known approach from OpenAI's #1196 solution. It repeatedly settled on a different argument, one the mathematician called cleaner, with "no analytic complications." Air-gapped, no internet, no information leakage.

GPT-5.5 solved numerous Erdős problems earlier this year. DeepMind's Nexus knocked out 9. Now Mythos, with a cleaner proof than the one that already existed.

Problems that survived 80 years are falling in weeks.

levent@__alpoge__

over the weekend i checked the obvious thing, which is whether mythos is able to solve the erdos unit distance problem, aka erdos problem #90. the answer is: yea

17d137.4K1.4K240

Sebastien Bubeck@SebastienBubeck

We now know that with an appropriate harness both Mythos and GPT-5.5 can reproduce what our internal model did in one-shot for the unit distance problem. Clearly there is an insane overhang of capabilities with this generation of models, and no ceiling in sight for what scientific advances they can bring. You can go and try to discover new things with 5.5 right now!

Xiao Ma@MaXiao54704

The standard GPT-5.5 reproduced the proof ~ 👇 https://chatgpt.com/share/6a0e9e04-8cb0-8332-a4f1-ec68acd2e03e

You don't need to wait for oai's internal model!

17d192.9K1.2K259

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.

23d108.1K894294

Pleometric@pleometric

GPT's Frightening Construction

thebes@voooooogel

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

23d81.8K859295

Digg Deeper

Sheing Ngasked

What's Erdős Unit Distance Conjecture?

The Erdős unit distance conjecture, posed by Paul Erdős in 1946, asks how many pairs at exact distance 1 can occur among n points in the plane.

Mathematicians long expected the maximum to be at most n^{1+o(1)}, matching square-grid constructions that achieve roughly n^{1 + c / log log n} unit distances. ^¹

An OpenAI model recently disproved it with new constructions yielding a fixed polynomial improvement.

Justin Mezzellasked

Can you ELI5 the Erdős Unit Distance Conjecture?

The Erdős unit distance conjecture asks: with n points scattered on a flat plane, what's the maximum number of pairs you can make exactly one unit apart? Erdős figured square grids achieve the most—roughly growing like n to the 4/3 power—without allowing way more repeats.