Examples for the use of AI and especially LLMs in notable mathematical developments

A counterexample to Erdős's unit distance conjecture has been found by an internal OpenAI model. This is, to date, by far the most famous problem solved by an AI.

An exposition of the proof (along with commentary from some prominent mathematicians) is available here.

OpenAI has produced a short marketing video. DISCLAIMER: The link is provided here for the convenience of those who think that OpenAI's statements are relevant to the discussion. It is not an endorsement of OpenAI's views or their products.

Boris Alexeev and Dustin Mixon posted last week their paper Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof, where they had an LLM generate the Lean formalization of their proof. In my view this is one of the promising uses of LLMs, because the verifier naturally guards against hallucinations.

The problem is notable: they give a counterexample to a $1000 Erdös problem (as well as noting that Marshall Hall had published a counterexample before Erdös made the conjecture).

My caveat: a human must still verify that the definitions and the statement of the main theorem are correct, lest the LLM generate a correct proof, but of a different theorem.

Here is an example Counterexample to majority optimality in NICD with erasures

From the abstract:

We asked GPT-5 Pro to look for counterexamples among a public list of open problems (the Simons ``Real Analysis in Computer Science'' collection). After several numerical experiments, it suggested a counterexample for the Non-Interactive Correlation Distillation (NICD) with erasures question: namely, a Boolean function on 5 bits that achieves a strictly larger value of E|f(z)| than the 5-bit majority function when the erasure parameter is p=0.40. In this very short note we record the finding, state the problem precisely, give the explicit function, and verify the computation step by step by hand so that it can be checked without a computer. In addition, we show that for each fixed odd n the majority is optimal (among unbiased Boolean functions) in a neighborhood of p=0. We view this as a little spark of an AI contribution in Theoretical Computer Science: while modern Large Language Models (LLMs) often assist with literature and numerics, here a concrete finite counterexample emerged.

This paper

Sergey Avvakumov, Roman Karasev, Tensor rank of the determinant and periodic triangulations of $\mathbb{R}^n$

https://arxiv.org/abs/2509.22333

includes in the Acknowledgments "We also thank ChatGPT 5 for pointing out that the lower bound in the proof of Theorem 1.5 can be stated in tensor language and is thus equal to the determinant’s tensor rank."

Erdős problem 1196 was very recently solved, autonomously, by GPT 5.4. See that link to erdosproblems.com for the details. In my view, this is the most dramatic advance in mathematics by an LLM to date. This was not an obscure problem; it was considered seriously by many excellent mathematicians. And GPT's solution involves a genuinely new perspective, which is likely to be applicable in more contexts in analytic number theory. Perhaps some of the commentators there, like Terry Tao and Will Sawin, who also frequently contribute to MO, can say more.

EDIT: These results now appear in this article on the arXiv - https://arxiv.org/abs/2605.00301

And here is a blog post from Terry Tao: https://terrytao.wordpress.com/2026/05/03/primitive-sets-and-von-mangoldt-chains-erdos-problem-1196-and-beyond/

At the request of Gil Kalai, I'm converting a comment to an answer.

The paper "Mathematical exploration and discovery at scale" by Bogdan Georgiev, Javier Gómez-Serrano, Terence Tao, and Adam Zsolt Wagner was just posted to the arXiv: https://arxiv.org/abs/2511.02864.

Below is the abstract of the paper.

AlphaEvolve is a generic evolutionary coding agent that combines the generative capabilities of LLMs with automated evaluation in an iterative evolutionary framework that proposes, tests, and refines algorithmic solutions to challenging scientific and practical problems. In this paper we showcase AlphaEvolve as a tool for autonomously discovering novel mathematical constructions and advancing our understanding of long-standing open problems. To demonstrate its breadth, we considered a list of 67 problems spanning mathematical analysis, combinatorics, geometry, and number theory. The system rediscovered the best known solutions in most of the cases and discovered improved solutions in several. In some instances, AlphaEvolve is also able to generalize results for a finite number of input values into a formula valid for all input values. Furthermore, we are able to combine this methodology with Deep Think and AlphaProof in a broader framework where the additional proof-assistants and reasoning systems provide automated proof generation and further mathematical insights. These results demonstrate that large language model-guided evolutionary search can autonomously discover mathematical constructions that complement human intuition, at times matching or even improving the best known results, highlighting the potential for significant new ways of interaction between mathematicians and AI systems. We present AlphaEvolve as a powerful new tool for mathematical discovery, capable of exploring vast search spaces to solve complex optimization problems at scale, often with significantly reduced requirements on preparation and computation time.

EDIT:

Some further developments are in the paper "New Nikodym set constructions over finite fields" by Terence Tao (https://arxiv.org/abs/2511.07721) whose abstract reads

For any fixed dimension $d \geq 3$ we construct a Nikodym set in $\mathbb{F}_q^d$ of cardinality $q^d - (\frac{d-2}{\log 2} +1+o(1)) q^{d-1} \log q$ in the limit $q \to \infty$, when $q$ is an odd prime power. This improves upon the naive random construction, which gives a set of cardinality $q^d - (d-1+o(1)) q^{d-1} \log q$, and is new in the regime where $\mathbb{F}_q$ has unbounded characteristic and $q$ not a perfect square. While the final proofs are completely human generated, the initial ideas of the construction were inspired by output from the tools AlphaEvolve and DeepThink. We also give a new construction of Nikodym sets in $\mathbb{F}_q^2$ for $q$ a perfect square that match the existing bounds of $q^2 - q^{3/2} + O(q \log q)$, assuming that $q$ is not the square of a prime $p \equiv 3 \pmod{4}$.

And also in the paper "Sum-difference exponents for boundedly many slopes, and rational complexity" again by Terence Tao (https://arxiv.org/abs/2511.15135) whose abstract reads

The dimension of Kakeya sets can be bounded using sum-difference exponents $\mathrm{SD}(R;s)$ for various sets of rational slopes $R$ and output slope $s$; the arithmetic Kakeya conjecture, which implies the Kakeya conjecture in all dimensions, asserts that the infimum of such exponents is $1$. The best upper bound on this infimum currently is $1.67513\dots$. In this note, inspired by numerical explorations from the tool AlphaEvolve, we study the regime where the cardinality of the set of slopes $R$ is bounded. In this regime, we establish that these exponents converge to $2$ at a rate controlled by the rational complexity of $s$ relative to $R$, which measures how efficiently $s$ can be expressed as a rational combination of slopes in $R$.

The "notability" here may be less because of the problem itself, and more because Donald Knuth was involved. Knuth wrote:

Shock! Shock! I learned yesterday that an open problem I’d been working on for several weeks had just been solved by Claude Opus 4.6— Anthropic’s hybrid reasoning model that had been released three weeks earlier! It seems that I’ll have to revise my opinions about “generative AI” one of these days. What a joy it is to learn not only that my conjecture has a nice solution but also to celebrate this dramatic advance in automatic deduction and creative problem solving. I’ll try to tell the story briefly in this note.

Here’s the problem, which came up while I was writing about directed Hamiltonian cycles for a future volume of The Art of Computer Programming:

Consider the digraph with $m^3$ vertices $ijk$ for $0 \le i, j, k < m$, and three arcs from each vertex, namely to $i^+jk$, $ij^+k$, and $ijk^+$, where $i^+ = (i+1) \bmod m$. Try to find a general decomposition of the arcs into three directed $m^3$-cycles, for all $m > 2$.

Knuth then went on to to describe how Filip Stappers used Claude interactively to discover a conjectural solution, which was then rigorously proved.

This does not seem to be widely circulating (not by one of the big tech companies): the lower bounds of the kissing numbers in dimensions 25 through 31 have been broken, using a two-player game-theoretic/reinforcement-learning approach.

https://arxiv.org/abs/2511.13391

The table on wikipedia and the one maintained by Henry Cohn have been updated.

(If I'm not mistaken, kissing number problems are also among that DeepMind's AlphaEvolve had attempted, breaking the 11d record (by 1); someone more familiar may provide the details.)

The paper “Point Convergence of Nesterov's Accelerated Gradient Method: An AI-Assisted Proof” by Uijeong Jang and Ernest Ryu, posted to Arxiv October 27, 2025, states in the abstract:

The Nesterov accelerated gradient method, introduced in 1983, has been a cornerstone of optimization theory and practice. Yet the question of its point convergence had remained open. In this work, we resolve this longstanding open problem in the affirmative. The discovery of the proof was heavily assisted by ChatGPT, a proprietary large language model, and we describe the process through which its assistance was elicited.

https://arxiv.org/abs/2510.23513

See also this discussion by Damek Davis that helps put the result in perspective: https://x.com/damekdavis/status/1982529760505782510?s=46

Following Sam Hopkins's suggestion, I'm posting an answer about First Proof (pun explanation here). This was a list of ten lemmas that professional mathematicians had proved in the normal course of their own research, but which they had not revealed in any public venue. The lemmas were posted to the arXiv in February 2026, with the proofs being withheld for about a week. Though First Proof was not a formal benchmark, the authors explicitly challenged AI models to prove the lemmas autonomously. Different models had varying degrees of success; you can get some sense of the state of the art from the report on Aletheia's performance. Although the models certainly did not knock all the problems out of the park, they performed better than most mathematicians had expected. I recommend reading Daniel Litt's lengthy blog post Mathematics in the Library of Babel. Litt had previously been somewhat skeptical of the capability of LLMs to do research-level math, but has updated his views in light of First Proof.

There is a second round in the works, which will unfold over the next few months and which will be more of a formal benchmark. In order to ensure, as far as possible, that the models really are run autonomously, the plan is for the First Proof team to run the models themselves, rather than simply trust AI companies to do so.

I recommend that anyone interested in First Proof listen to this interview with Dan Spielman and Nikhil Srivastava, during which they explain the motivation behind First Proof and what they hope to achieve in the future. (Also, there was a live event at MoMath about First Proof on Pi Day, but I don't know if there is a publicly available video of that event.)

EDIT: The second batch results are in. To oversimplify a bit, this second batch was a test of the best publicly available (for a fee) ChatGPT and Gemini models, when used with an open-source harness. Depending on how stringent your definition of "solved" is, with a 24-hour total time budget, GPT 5.5 Pro solved between 3 and 7 of the problems, while Gemini 3.1 Pro Preview solved at most 1 of the problems. While the sample size is of course small, this is to my knowledge the best benchmark out there right now for assessing how well the publicly available models can be expected to perform on a "random research-level lemma."

Scott Aaronson Phillip Harris, Freek Witteveen have a recent paper on the bounds of amplification of QMA (quantum Merlin-Arthur). A critical part of the paper involved a linear algebra trick suggested by GPT5. See Aaronson's blog entry here.

On 2026-05-27, a variant of Erdős problem #52 (Erdős' similar conjecture for sets of reals, distinct from the conjecture for sets of integers presented as primary by erdosproblems.com, but also discussed there under problem #52)

For $A$ a finite set of real numbers, the sets $A+A$ of pairwise sums and $A\cdot A$ of pairwise products have $\max(|A+A|,|A\cdot A|)\ge|A|^{2-o(1)}$

was disproven by Bloom, Sawin, Schildkraut, and Zhelezov. in The sum-product conjecture is false for real numbers, who constructed a counterexample with an exponent on the upper bound a tiny bit below 2. They also were able to obtain $\le|A|^{1.906}$ for a specific function field case.

It ends with a disclaimer:

The role of AI in this proof. The authors were inspired to revisit the possibility of disproving the sum-product conjecture using number fields of large degree by the recent OpenAI counterexample to the unit distance conjecture (see [2]). Curiously, the final construction given here required far less number theoretic input than the unit distance counterexample. GPT-5.5 Pro was used as a sounding board in the early stages of the development of this proof, but the final proof, including all the main ideas, was almost entirely human-generated (the exception being the suggestion of Lemma 3.4, which replaced a more complicated result of Schinzel with a short elementary argument). Everything in this paper was written by the authors.

Not exactly a notable result, but in my recent preprint Evaluation of GPT-5 on an Advanced Extension of Kashihara's Problem I describe how GPT-5 has been able to improve the general version of an extended combinatorial problem I originally solved in 2010.

The abstract of Early science acceleration experiments with GPT-5 by Sébastien Bubeck, Christian Coester, Ronen Eldan, Timothy Gowers, Yin Tat Lee, Alexandru Lupsasca, Mehtaab Sawhney, Robert Scherrer, Mark Sellke, Brian K. Spears, Derya Unutmaz, Kevin Weil, Steven Yin, and Nikita Zhivotovskiy states in part, "Of note, this paper includes four new results in mathematics (carefully verified by the human authors), underscoring how GPT-5 can help human mathematicians settle previously unsolved problems."

There have been some notable recent examples of LLMs playing an important role in solving certain Erdős problems, e.g., Problem 124 and Problem 481 (although I think the latter turned out to be implied by a result of Klarner in 1982), which were purportedly solved entirely by Aristotle, and Problem 367, which was reduced by Wouter van Doorn to a lemma that was solved by Gemini Deepthink. While these are not famous Erdős problems and turned out to have relatively short and simple solutions, they differ from Olympiad problems in that the computer solved problems without known solutions.

This is a modest development and a modest use of AI, but nonetheless, since it may seem that only discretish mathematics is mentioned in most answers, the proof of Lemma 6 in https://arxiv.org/pdf/2511.06849 is due to ChatGPT 5. The techniques are standard but their use is elegant. Honestly, we were stressed out about the proof (when we discovered that an earlier one we had was flawed), and ChatGPT came to our rescue.

here is a paper on bottleneck duality in flow networks with lattice coefficients from fall 2024.

https://arxiv.org/abs/2410.00315

the appendix to this paper details how the main result and the proof were generated by GPT-o1-mini in september 2024. it was very difficult to get a correct proof at the time; current models nail a correct proof immediately.

I have hesitated to post this example because I don't think it's really a "notable mathematical development" as such, but after seeing the other answers, I think this one is worth mentioning.

As reported in Scientific American, Epoch AI invited several mathematicians, including Ken Ono, to a meeting designed to generate challenge problems for "FrontierMath". Among other things, Ono came up with what he thought was a Ph.D.-thesis-level problem: "What is the 5th power moment of Tamagawa numbers of elliptic curves over $\mathbb{Q}$?" To Ono's amazement, the AI autonomously solved the problem. You can read Ono's account on his Facebook page (also reproduced below), or listen to him talk about it here.

Even if this is a cherry-picked example—the best one from the whole meeting—this strikes me as a very impressive achievement. But see also this tweet by Daniel Litt, who was also one of the invited mathematicians but was not too impressed when he read over the chat log.

A few days ago, improving upon a result of Klartag, the authors Abuya (pseudonym), Gargava and Zhao released this preprint here: https://arxiv.org/pdf/2606.05105. They showed that there exists a universal constant $c>0$ and an infinite sequence of dimensions $N$ such that the lattice sphere packing density in dimension $N$ can be bounded as follows: $$\delta_N^{\textrm{lat}} \ge cN^2\log \log N2^{-N}.$$ The previous bound held by Klartag was $cN^22^{-N}$. The prompts and responses on GPT 5.5 Pro are available in the ancillary files on arxiv just under the tex source of the preprint.

Using deep neural networks, Deepmind & collaborators numerically found a class of unstable singularities of the porous media and 3D Euler (with boundary) equations. Notable here is the fact that the level of precision of their solutions "meets the stringent requirements for rigorous mathematical validation via computer-assisted proofs" (quote from the paper).

Paper here: https://arxiv.org/abs/2509.14185 Article: https://deepmind.google/discover/blog/discovering-new-solutions-to-century-old-problems-in-fluid-dynamics/

(I know that there was some discussion of whether to keep answering this question with new examples, but this one seemed cool enough to me to include here.)

In "Bruhat intervals that are large hypercubes" (https://arxiv.org/abs/2601.01235), Jordan Ellenberg, Nicolas Libedinsky, David Plaza, José Simental, and Geordie Williamson find large hypercubes (of order $n \log n$, in particular, superlinear) inside of Bruhat graphs of the symmetric group $S_n$, a problem of interest for various connections including to cluster algebras of Richardson varieties and the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials. They used the AlphaEvolve AI system to find these intervals - more specifically, they studied examples of algorithms based on data for small $n$, and then were able to give a natural "interpretation" to what the machine was doing (at least in the case of $n=2^m$ a power of two) to find a natural class of permutations (the "dyadically well-distributed" ones) which form a big hypercube interval in the Bruhat graph.

The abstract of their paper is

We study the question of finding big Bruhat intervals that are poset hypercubes in the symmetric group $S_n$. Using permutations suggested by AlphaEvolve (an evolutionary coding agent developed by Google DeepMind), we were led to an unusual situation in which the agent produced a pattern which performed well for the $n$ tested, and which we show works well for general $n$. When $n$ is a power of 2 we exhibit a hypercube of dimension $O(n\log n)$, matching the largest possible dimension up to a constant multiple. Furthermore, we give an exact characterization of the vertices of this hypercube: they are precisely the dyadically well-distributed permutations, a simple digitwise property that already appeared in connection with Monte Carlo integration and mathematical finance. The maximal dimension of a Bruhat interval that is an hypercube in $S_n$ gives a lower bound (and possibly is equal to) the maximal possible coefficient of the second-highest degree term in the Kazhdan--Lusztig $R$-polynomial in $S_n$. As a surprising consequence, we obtain a new lower bound of order $n\log n$ for the maximal number of frozen variables appearing in the cluster algebras attached to the open Richardson varieties in $S_n$, and a similar result for moduli spaces of embeddings of Bruhat graphs.

I initiated a meta.MO discussion on whether it's worthwhile to continue posting answers here. Denis T suggested that a minimum threshold would be "a defendable case that the same result could not be achieved by using Google a few times." Ravi Vakil commented that he has definitely seen such examples, so I think it is worth posting a link to the following paper, which was co-authored by Vakil.

The abstract of The motivic class of the space of genus 0 maps to the flag variety by Jim Bryan, Balázs Elek, Freddie Manners, George Salafatinos, and Ravi Vakil says in part, "The proof of this result was obtained in conjunction with Google Gemini and related tools. We briefly discuss this research interaction, which may be of independent interest. However, the treatment in this paper is entirely human-authored (aside from excerpts in an appendix which are clearly marked as such)."

A recent paper by Nagda, Raghavan, and Thakurta: "Reinforced generation of combinatorial structures: Applications to complexity theory" They received help from AlphaEvolve to improve the best-known bound for Max-3CUT and Max-4CUT. Their idea seems quite general, so I would not be surprised if more complexity theory results would be improved.

The computational complexity paper "Search versus Decision for $S_2^P$" by Lance Fortnow writes in the acknowledgements:

While the results are fully due to the author, this paper was mostly generated using the large language model Gemini 3 Pro with prompting from the author. The author takes full responsibility for its contents.

EDIT (additional context): The author further elaborated on Twitter/X: when asked "It looks like you only told it the theorem statement and didn't give it the sketch." the author replied "Yes, it came up with the proof on its own. Surprised me as well.".

The paper Solving a Research Problem in Mathematical Statistics with AI Assistance by Edgar Dobriban documents how GPT-5 Pro helped close a gap between upper and lower bounds in robust density estimation under Wasserstein contaminations.

The mathematical results appear in version 3 of Minimax Statistical Estimation under Wasserstein Contamination (Chao–Dobriban). Notably, the math paper itself contains no mention of AI assistance—the role of GPT-5 is documented only in the separate meta-paper.

The key AI contributions were:

• For the upper bound: suggesting a sharper analysis of the contamination bias via the optimal transport map and interpolation along geodesics
• For the lower bound: proposing the dynamic Benamou–Brenier formulation of optimal transport, a technique the authors were unfamiliar with

The meta-paper provides detailed transcripts and a balanced discussion of limitations (hallucinated references, glossed-over details requiring days to verify).

See also the author's Twitter/X thread summarizing the experience.