OpenAI solves 80-year Erdős geometry problem: AI autonomously disproves the square grid conjecture (May 2026)

On May 20, 2026, OpenAI announced that an internal reasoning model has independently solved the planar unit distance problem—an 80-year-old open question first posed by Paul Erdős in 1946. The problem asks: if you place n points in a plane, what is the maximum number of pairs that can be exactly distance 1 apart? For nearly eight decades, mathematicians believed square grids were essentially optimal. The OpenAI model disproved this longstanding conjecture, discovering an infinite family of constructions using deep algebraic number theory (Golod-Shafarevich theory, infinite class field towers) that achieve polynomial improvement over square grids—specifically, n^(1+δ) unit-distance pairs for some fixed δ > 0 (later refined to δ = 0.014 by Princeton mathematician Will Sawin).

This marks the first time AI has autonomously solved a prominent open problem central to a subfield of mathematics. The proof was not guided step-by-step by humans, nor did it complete a partial proof—the model received the problem statement and produced the solution independently. Fields medalist Tim Gowers calls the result "a milestone in AI mathematics," and Noga Alon (a leading combinatorialist at Princeton) describes the solution as "an outstanding achievement" that "applies fairly sophisticated tools from algebraic number theory in an elegant and clever way."

This article covers what the problem is, what the AI proved, how the proof works, expert reactions, and what this means for mathematics and AI.

TL;DR

Primary source: OpenAI blog · 125-page proof PDF · Companion paper by mathematicians

The Planar Unit Distance Problem: What Erdős Asked

In 1946, Hungarian mathematician Paul Erdős posed a deceptively simple question:

If you place n points in the plane, what is the maximum number of pairs that can be exactly distance 1 apart?

Notation: Let u(n) = the maximum number of unit-distance pairs among n points.

Why it's hard:

• Easy to state (anyone can understand it)
• Remarkably difficult to resolve (80 years of research)
• Research Problems in Discrete Geometry (2005 book) calls it "possibly the best known (and simplest to explain) problem in combinatorial geometry"
• Noga Alon (Princeton): "one of Erdős's favorite problems"
• Erdős offered a monetary prize for resolving it

What Was Known Before: Square Grids Seemed Optimal

Simple constructions

Linear growth is easy:

• Place n points in a line → n-1 pairs
• Place points in a square grid → ~2n pairs

The best known construction (pre-2026)

A rescaled square grid gives n^(1+C/loglog(n)) pairs for some constant C.

Key insight: Since loglog(n) tends to infinity with n, the exponent 1 + C/loglog(n) tends to 1, meaning growth is only slightly faster than linear.

Erdős's conjecture

Erdős conjectured an upper bound of n^(1+o(1)), where o(1) indicates a term tending to 0 as n→∞.

Translation: The growth rate should be essentially linear—square grids are nearly optimal, and no construction can achieve polynomial improvement (e.g., n^1.01 or n^1.1).

Upper bounds

The best known upper bound is O(n^(4/3)) (Spencer, Szemerédi, Trotter, 1984), refined by others but essentially unchanged for 40+ years.

The gap:

• Lower bound (constructions): ~n^(1+o(1))
• Upper bound (proof): O(n^(4/3))

For 80 years, the belief was that the lower bound was tight—square grids were optimal.

What the OpenAI Model Proved: Square Grids Are Not Optimal

The OpenAI model disproved Erdős's conjecture by constructing configurations with:

n^(1+δ) unit-distance pairs for some fixed exponent δ > 0

What this means:

• δ > 0 is a constant (not tending to 0)
• The exponent 1 + δ is strictly greater than 1
• This is a polynomial improvement over square grids

Explicit value: The original AI proof did not compute δ explicitly, but Will Sawin (Princeton professor) refined the construction and showed:

δ = 0.014

Translation: For infinitely many values of n, you can place n points with at least n^1.014 unit-distance pairs.

Why this matters:

• Disproves 80-year conjecture (Erdős's n^(1+o(1)) upper bound is false)
• Square grids are not optimal (there exist fundamentally better constructions)
• Unexpected mathematical connection (algebraic number theory → discrete geometry)

How the AI Solved It: Algebraic Number Theory to the Rescue

The proof uses sophisticated tools from algebraic number theory that were well-known to specialists but had never been connected to geometric questions in the Euclidean plane.

Starting point: Gaussian integers

Erdős's original lower bound used the Gaussian integers: numbers of the form a + bi, where:

• a, b are integers
• i is the square root of -1

Why Gaussian integers help:

• They form a grid in the complex plane
• They have unique factorization (like ordinary integers)
• Distance formulas simplify nicely

Erdős's insight: Place points at Gaussian integers → many unit-distance pairs.

The AI's breakthrough: Richer number fields

The AI replaced Gaussian integers with more complicated generalizations from algebraic number theory—specifically:

1. Algebraic number fields (extensions of ordinary integers/rationals)
2. Infinite class field towers (sequences of field extensions with special properties)
3. Golod-Shafarevich theory (tool to prove these towers exist)

Why richer fields help:

• They have more symmetries than Gaussian integers
• These symmetries create more unit-length differences
• The construction achieves n^(1+δ) instead of n^(1+o(1))

Technical machinery: The proof uses:

• Infinite class field towers (exist by Golod-Shafarevich)
• Factorization properties in number fields
• Embedding number fields into the plane (viewing algebraic numbers as geometric points)

Quote from external mathematicians:

"The solution... applies fairly sophisticated tools from algebraic number theory in an elegant and clever way. These ideas were well-known to algebraic number theorists, but it came as a great surprise that these concepts have implications for geometric questions in the Euclidean plane."

The Proof: 125 Pages of Chain-of-Thought Reasoning

Proof structure:

• Main proof: 125-page PDF
• Abridged chain of thought: Available on OpenAI's site
• Companion paper: Written by external mathematicians (Gowers, Alon, Shankar, Tsimerman)

How the model reasoned:

1. Explored Gaussian integer approach (Erdős's original method)
2. Generalized to algebraic number fields (looking for richer structure)
3. Discovered connection to class field towers (via Golod-Shafarevich)
4. Constructed explicit configurations (placed points using number-theoretic symmetries)
5. Proved polynomial improvement (counted unit-distance pairs, showed n^(1+δ))

Quote from OpenAI:

"The proof brings unexpected, sophisticated ideas from algebraic number theory to bear on an elementary geometric question."

Expert Reactions: "A Milestone in AI Mathematics"

Tim Gowers (Fields medalist, Cambridge):

"This has been one of Erdős' favorite problems... Every mathematician working in Combinatorial Geometry thought about this problem... The solution by the internal model of OpenAI is, in my opinion, an outstanding achievement, settling a long-standing open problem. The fact that the correct answer is not n^(1+o(1)) is surprising."

"In my opinion, this is a milestone in AI mathematics."

Noga Alon (Princeton, leading combinatorialist):

"This has been one of Erdős' favorite problems... The solution... is an outstanding achievement... The construction and its analysis apply fairly sophisticated tools from algebraic number theory in an elegant and clever way."

Arul Shankar (number theorist):

"In my opinion this paper demonstrates that current AI models go beyond just helpers to human mathematicians – they are capable of having original ingenious ideas, and then carrying them out to fruition."

Thomas Bloom (companion paper author):

"When assessing the importance of an AI-generated proof, I ask: has this taught us something new? Do we understand discrete geometry better now? I think the answer is a moderated yes: this shows that number theoretic constructions have a lot more to say about these questions than we suspected; moreover, that the number theory required can be very deep."

Jacob Tsimerman:

"The solution demonstrates depth of reasoning and the ability to connect distant areas of mathematics."

What This Means for Mathematics

1. AI can solve frontier research problems

This is not a textbook problem or a known result. It's a prominent open problem that professional mathematicians had worked on for 80 years.

Significance: AI systems are now capable of original research at the frontier.

2. Unexpected connections revealed

The proof links algebraic number theory (abstract, algebraic) to discrete geometry (concrete, visual).

Quote from Bloom:

"This shows that there is a lot more that number theoretic constructions have to say about these sorts of questions than we suspected... No doubt many algebraic number theorists will be taking a close look at other open problems in discrete geometry in the coming months."

Translation: The AI discovered a bridge between two fields, potentially opening new research directions for human mathematicians.

3. Human expertise still essential

The AI produced a 125-page proof, but external mathematicians checked it, explained it, and provided context.

Companion paper contribution:

• Validated correctness
• Explained significance
• Connected to broader literature
• Suggested future directions

Bloom's view:

"AI is helping us to more fully explore the cathedral of mathematics we have built over the centuries; what other unseen wonders are waiting in the wings?"

What This Means for AI

1. General-purpose reasoning models are capable

The proof came from a general-purpose reasoning model, not a system:

• Trained specifically for mathematics
• Designed for this problem
• Given hints or partial proofs

OpenAI's statement:

"The proof came from a new general-purpose reasoning model, rather than from a system trained specifically for mathematics... It produced a proof resolving the open problem."

Significance: The same reasoning capabilities that solve math problems can potentially apply to science, engineering, and AI research itself.

2. Depth of reasoning demonstrated

Mathematics is a clear testbed for reasoning:

• Problems are precise
• Proofs can be checked
• Long arguments only work if reasoning holds from beginning to end

Quote from OpenAI:

"Mathematics provides a particularly clear testbed for reasoning... a long argument only works if the reasoning holds together from beginning to end."

The 125-page proof shows the model can maintain coherent reasoning over extended chains of thought.

3. Self-improving AI research possible

OpenAI's broader vision:

"AI is about to start taking a very serious role in the creative parts of research, and most importantly AI research itself."

Translation: If AI can solve frontier math problems, it can potentially:

• Suggest better training algorithms
• Design improved architectures
• Optimize hyperparameters
• Debug training runs

This connects to Andrej Karpathy's move to Anthropic (see our Karpathy blog post)—using AI to accelerate AI development.

The Model: What We Know (and Don't Know)

What OpenAI disclosed:

• Internal reasoning model (not publicly available)
• General-purpose (not math-specific)
• Tested on a collection of Erdős problems (part of broader research evaluation)
• Produced 125-page proof (available for download)

What OpenAI did NOT disclose:

• Model name (o3? o4? Something new?)
• Training data (math-specific corpora? General web + textbooks?)
• Inference-time compute (how many forward passes? Cost per proof?)
• Success rate (did it solve other Erdős problems? What percentage?)

Likely architecture:

Based on OpenAI's recent releases (o1, o3), the model probably uses:

• Chain-of-thought reasoning (extended internal monologue before answering)
• Reinforcement learning (trained to maximize correct proofs on math problems)
• Inference-time scaling (more compute → better reasoning)

Limitations and Open Questions

1. Peer review pending

The proof has been checked by external mathematicians (Gowers, Alon, Shankar, Tsimerman) and they wrote a companion paper, but:

• Full peer review in a mathematics journal is still pending
• The community needs time to digest the 125-page proof
• Subtle errors could still exist

Status: Highly credible (Fields medalist + top combinatorialists validated it), but not yet published in a journal.

2. Generalization unclear

Questions:

• Can the model solve other open problems at this level?
• Was this a one-off success, or is frontier research now routine for AI?
• Success rate on Erdős problems not disclosed

What we need: More examples of AI solving different types of open problems.

3. Human guidance still needed

The model autonomously produced the proof, but:

• Humans selected the problem
• Humans checked the proof
• Humans wrote the companion paper explaining significance

Quote from OpenAI:

"That future still depends on human judgment. Expertise becomes more valuable, not less. AI can help search, suggest, and verify. People choose the problems that matter, interpret the results, and decide what questions to pursue next."

How This Compares to Previous AI Math Milestones

Key difference: This is not a competition problem or a known result—it's a professional research problem that stumped experts for 80 years.

What Happens Next

Short term (2026):

• Peer review of the 125-page proof
• Community verification by discrete geometers and number theorists
• Refinements (e.g., Will Sawin's δ = 0.014 improvement)

Medium term (2027-2028):

• More AI-solved open problems (other Erdős problems, other fields)
• Human-AI collaboration on frontier research
• New research directions (algebraic number theory → discrete geometry)

Long term (2029+):

• AI research assistants become standard in mathematics
• Automated conjecture generation (AI proposes new conjectures based on patterns)
• Self-improving AI uses similar reasoning to accelerate its own development

Related on ExplainX

• Andrej Karpathy joins Anthropic to use AI for AI research
• o1 and o3: OpenAI's reasoning models explained
• What are agent skills? Complete guide
• AlphaGeometry: Google DeepMind's IMO geometry solver
• Scaling laws: why bigger models aren't always better

Sources

• OpenAI blog — Model disproves discrete geometry conjecture: openai.com
• 125-page proof PDF: cdn.openai.com
• Companion paper by mathematicians: Available via OpenAI blog
• Interesting Engineering — 80-year mystery cracked: interestingengineering.com
• CryptoBriefing — OpenAI solves unit distance problem: cryptobriefing.com
• Digg — OpenAI internal model breakthrough: digg.com

Mathematical proofs require peer review and community validation. Treat this as May 22, 2026 context—the proof has been checked by leading mathematicians (Gowers, Alon, Shankar, Tsimerman) but full journal peer review is pending. Verify claims against primary sources before citing in academic work.