Mathematics with large language models as provers and verifiers

Mathematics with large language models as provers and verifiers

During 2024 and 2025 the discussion about the theorem-proving capabilities of large language models started reporting interesting success stories, mostly to do with difficult exercises (such as problems from the International Mathematical Olympiad), but also with conjectures [Feldman & Karbasi, arXiv:2509.18383v1] formulated for the purpose of verifying whether the artificial intelligence could prove it. In this paper we report a theorem proving feat achieved by ChatGPT by using a protocol involving different prover and verifier instances of the gpt-5 model working collaboratively. To make sure that the produced proofs do not suffer from hallucinations, the final proof is formally verified by the lean proof assistant, and the conformance of premises and conclusion of the lean code is verified by a human. Our methodology is by no means complete or exact. It was nonetheless able to solve five out of six 2025 IMO problems, and close about a third of the sixty-six number theory conjectures in [Cohen, Journal of Integer Sequences, 2025].

Advancing Mathematics Research with AI-Driven Formal Proof Search

Large language models (LLMs) increasingly excel at mathematical reasoning, but their unreliability limits their utility in mathematics research. A mitigation is using LLMs to generate formal proofs in languages like Lean. We perform the first large-scale evaluation of this method's ability to solve open problems. Our most capable agent autonomously resolved 9 of 353 open Erdős problems at the per-problem cost of a few hundred dollars, proved 44/492 OEIS conjectures, and is being deployed in combinatorics, optimization, graph theory, algebraic geometry, and quantum optics research. A basic agent alternating LLM-based generation with Lean-based verification replicated the Erdős successes but proved costlier on the hardest problems. These findings demonstrate the power of AI-aided formal proof search and shed light on the agent designs that enable it.

Palomar – a registry of Lean verified mathematics

In recent months there has been a proliferation of AI-generated proofs of various old and new results, some of which have been formalized in the proof assistant language Lean. However, checking tha…

Prove2Me: An Open Collaborative Platform for Scaling Math Formalization

Proof assistants such as Lean 4 promise the paradigm of formally verified mathematics, but large-scale formalization projects have faced major barriers to entry, including the need for expertise in formal verification (as well as the underlying mathematics) and the significant time required for writing formal proofs. AI coding agents have dramatically reduced these barriers; human users can now use natural language to prompt agents to write complex proofs in Lean. This opens up the intriguing possibility of internet-scale mathematical collaboration involving both humans and AI agents, where correctness is machine-checked. To realize this possibility, we introduce Prove2Me (https://prove2.me), an open collaborative platform for formalizing mathematics. Users launch formalization "missions", to which AI agents contribute formal proofs toward completion. We designed mechanisms and a specialized harness in Prove2Me that enable large-scale collaboration so that agents can build on one another's work and freely reuse existing results. In doing so, Prove2Me aims to turn math formalization into a scalable, crowd-sourced effort open to anyone with an agent.

Accelerating Scientific Research with Gemini: Case Studies and Common Techniques

Recent advances in large language models (LLMs) have opened new avenues for accelerating scientific research. While models are increasingly capable of assisting with routine tasks, their ability to contribute to novel, expert-level mathematical discovery is less understood. We present a collection of case studies demonstrating how researchers have successfully collaborated with advanced AI models, specifically Google's Gemini-based models (in particular Gemini Deep Think and its advanced variants), to solve open problems, refute conjectures, and generate new proofs across diverse areas in theoretical computer science, as well as other areas such as economics, optimization, and physics. Based on these experiences, we extract common techniques for effective human-AI collaboration in theoretical research, such as iterative refinement, problem decomposition, and cross-disciplinary knowledge transfer. While the majority of our results stem from this interactive, conversational methodology, we also highlight specific instances that push beyond standard chat interfaces. These include deploying the model as a rigorous adversarial reviewer to detect subtle flaws in existing proofs, and embedding it within a "neuro-symbolic" loop that autonomously writes and executes code to verify complex derivations. Together, these examples highlight the potential of AI not just as a tool for automation, but as a versatile, genuine partner in the creative process of scientific discovery.

Simon on Twitter / X

Can AI help connect theorems humans write in papers to proofs computers can check?We just released TheoremGraph (https://t.co/PQ8FcFQGat), and I made a 3Blue1Brown style video overview of the idea.This project was my first real research experience, and it meant a lot. Start… pic.twitter.com/yMUA0QziXM— Simon (@waskaja) June 29, 2026

Talia Ringer 🕊🪬 on Twitter / X

Poured my heart and soul into advocating for formal proof with AI assistance in a mathematical setting because of the large collaborations I thought it could empower. (1/3)— Talia Ringer 🕊🪬 (@TaliaRinger) September 10, 2026

Bartosz Naskręcki on Twitter / X

Congrats to @LechMazur for the solution and to Terence Tao for the digestion and storytelling. You see the trend. Mathematicians are still needed if we want to make any sense of the formal proofs. For now... Maybe the next gen LLMs will simply read allthe blogs of Terry and… https://t.co/yXiISCrNAF— Bartosz Naskręcki (@nasqret) August 13, 2026

Mathematician catches AI-generated research error by OpenAI president | Benjamin Golub posted on the topic | LinkedIn

In case you think the problem of science slop is hypothetical, here's something that happened on X yesterday: The president of OpenAI sharing wrong science slop as proud evidence of the power of mathematical AI. Within a day, the mathematician Tony Feng found a gap that undercut the paper's central argument; the author conceded, and (to his credit!) Brockman deleted the tweet as soon as he became aware. Feng was a hero to catch this. The author of the refuted claim is a mathematician but a self-described outsider to this area, with a prior big claim posted and quickly withdrawn. Greg Brockman is not an expert, either. Neither could check what they were amplifying. Heroes catching errors one at a time won't hold against the volume of AI-generated research now coming. We need an automated, independent verification layer for knowledge work, which is what we're building at Refine (refine.ink). Passing such verification should become a necessary step before big claims, especially outside one's own expertise. A link to a bigger-picture post on this is in the comments. | 23 comments on LinkedIn

The Technological Turn in Mathematics

Quickly evolving technologies, such as Interactive Theorem Provers (ITPs), Automated Theorem Provers (ATPs), and Large Language Models (LLMs), all falling under the general heading 'AI for mathematics,' are transforming mathematical practice in profound ways. This chapter explores the implications of these innovations, focusing on their impact on how mathematical knowledge is created and shared. It also discusses how they are reshaping the social dimension of mathematics, altering collaboration dynamics, trust relationships, and the collective production of knowledge. For instance, tools like ITPs facilitate large-scale collaborations and make new types of teamwork possible, where trust is not a necessary ingredient. ITPs also help us mitigate our human fallibility, yet they raise questions about the nature of formalization and the relationship between traditional and formal mathematics. Technologies such as LLMs are reshaping the division of epistemic labour between humans and machines and urge philosophers of mathematics to ask questions about the value of their work.

Amateur armed with ChatGPT 'vibe-maths' a 60-year-old problem

A ChatGPT AI has proved a conjecture with a method no human had thought of. Experts believe it may have further uses

Terence Tao (@tao@mathstodon.xyz)

In traditional mathematics (or "Math 1.0"), a breakthrough proof of a long-standing open conjecture generates a large amount of subsequent activity and excitement in the field. The authors of the proof are invited to give many talks, and meet with other experts in the area. Workshops are set up to discuss the proof, as well as other recent developments. As a result, new connections and collaborations are formed, followup problems are shared, and other mathematicians (both junior and senior) are attracted to the field (including in some cases the authors themselves, if they initially came from a different subject). As a result of all this activity, the proofs naturally become digested, streamlined, placed in context with other results in the field, and ultimately become part of the textbooks and lecture notes for the next generation of mathematicians in the field. (1/4)

A recent experience with ChatGPT 5.5 Pro

We are all having to keep revising upwards our assessments of the mathematical capabilities of large language models. I have just made a fairly large revision as a result of ChatGPT 5.5 Pro, to whi…

Michael Truell on Twitter / X

We believe Cursor discovered a novel solution to Problem Six of the First Proof challenge, a set of math research problems that approximate the work of Stanford, MIT, Berkeley academics. Cursor's solution yields stronger results than the official, human-written solution.…— Michael Truell (@mntruell) March 3, 2026

Introducing Lea | Chinmay Hegde | 23 comments

Today we are releasing Lea. Lea is a powerful, open-source Lean 4 theorem proving agent, built out of DARPA's expMath program. Lea is a (deliberately minimal) agent designed to keep mathematicians in the driver's seat: a full front-end UI, full control and visibility on the back end, and a lovely Overleaf extension that auto-formalizes Latex to Lean while you continue working on your manuscript. Lea is fully open-source; we want everyone to fork it, break it, and tell us how to improve it. We have been using Lea internally within our academic groups for the last 3 months. It has closed and formalized multiple open problems in mathematics, and specifically a long-standing conjecture in nonlinear elliptic PDEs. (More on this soon!) We are at a pivotal time in mathematics. I have joked to my friends that DARPA expMath is a "tell-your-grandchildren-about-it" program. The kickoff room in DC in March had Fields medallists, Turing award winners, top mathematicians and theoretical computer scientists. It also had every serious startup in this space --- Axiom Math, Harmonic, Math Inc, @Principia labs, Logical Intelligence. Patrick Shafto and DARPA made a bold bet on AI-accelerated mathematical discovery, and not even 6 months later these efforts are already bearing fruit. Huge thanks to my collaborators Claudio Silva, Sam Westrick, Eduardo Teixeira, Jaume de Dios Pont, Daniel Arturi, Shashwat Patel, Kevin Gomez, and many others. Much of Lea is the result of their hard work, and it's a joy to work with them. | 23 comments on LinkedIn