Tom Hundley
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On May 21, 2026, a DeepMind-affiliated team posted arXiv preprint 2605.22763, reporting that an AI agent autonomously solved 9 of 353 open Erdős problems — two of which had been open for 56 years — and proved 44 of 492 open OEIS conjectures. Every result was Lean-verified. The cost was roughly a few hundred dollars per problem.
That combination is what makes the paper consequential. This is not another benchmark score or competition-math result. It is a credible demonstration of AI producing original, research-level mathematics, with credibility grounded in machine-checked proofs rather than persuasive prose. For technical leaders, the practical takeaway is straightforward: when an AI system can generate verifiable new results for a cloud-bill-scale cost, the question shifts from whether AI can reason at all to where verified machine reasoning becomes cheaper than expert human labor.
TL;DR: The paper reports 9 solved Erdős problems and 44 proved OEIS conjectures, all formally verified in Lean, at a cost of roughly a few hundred dollars per problem.
The preprint (arXiv:2605.22763), whose authors include Pushmeet Kohli, reports results across two established mathematical problem sets:
| Problem Set | Total Open Problems | AI-Solved | Notable Detail |
|---|---|---|---|
| Erdős problems | 353 | 9 | Two had been open for 56 years |
| OEIS conjectures | 492 | 44 | Results formally verified in Lean |
A few boundaries matter here. The paper's claims are scoped to these specific Erdős and OEIS problem sets. It does not claim broad superhuman reasoning, and it does not justify sweeping conclusions outside formal mathematics. It also does not identify the system using the product branding or engine names that appeared in some press coverage.
Paul Erdős left behind a large collection of open problems that became landmarks across combinatorics, number theory, and graph theory. Many are compact to state and difficult to solve, which is part of why they have remained active targets for mathematicians for decades.
OEIS conjectures are a different kind of challenge. They arise from observed patterns in integer sequences and require formal proof rather than empirical plausibility. They may be less famous than Erdős problems, but they are still rigorous open mathematical questions.
The raw counts are the headline, but the context matters. Solving 9 of 353 open Erdős problems is notable because these were not benchmark exercises with known answers. Two had remained open for 56 years. The 44 of 492 OEIS result is similarly meaningful because it reflects successful proof generation on genuinely open conjectures rather than recovery of known solutions.
TL;DR: Lean verification matters because it turns AI-generated proofs from plausible text into machine-checked mathematical results.
The central weakness of language-model reasoning in technical domains is familiar: outputs can look correct while hiding subtle logical gaps. In mathematics, that failure mode is fatal. A proof is either valid or it is not.
This is why Lean verification changes the significance of the result.
Lean is an interactive theorem prover and programming language. When a proof is Lean-verified, the proof has been checked mechanically by the Lean kernel.
In practice, that means:
That standard is stricter than ordinary prose review. Human experts can overlook a gap in a long argument. A formal proof checker does not accept the proof unless the logic closes.
For software teams, the analogy is close to compilation plus formal checking. A generated function that looks right is not enough; it has to run and satisfy constraints. Here, a generated proof that reads well is not enough; it has to pass a theorem prover.
That is why the paper lands differently from many AI reasoning demos. The interesting question is not whether the output sounds mathematical. The proofs are machine-checked. That makes the result useful as evidence of verified reasoning rather than polished text generation.
TL;DR: The breakthrough is not that AI solved hard math problems, but that it solved open ones with no existing human solution to copy.
AI systems have already shown strong performance on olympiad-style and benchmark mathematics. Those results mattered, but they had an important limitation: the problems were already solved by humans. The system's task was to recover or rediscover a known answer.
This paper is different.
Competition problems are designed to be solvable. They test ingenuity, but they come with the implicit guarantee that a correct solution exists and is already known. That makes them useful for evaluation, but limited as evidence of original discovery.
Open Erdős problems and OEIS conjectures are different by definition. There is no answer key. A successful proof extends the mathematical record.
In this context, original research means the system produced proofs for statements that were previously unresolved within the targeted sets. Before this work, those problems were open. After it, some were no longer open.
That is the key threshold crossed by arXiv:2605.22763. The result is not better benchmark performance. It is the production of new, formally verified mathematical knowledge within the scope of the paper.
TL;DR: The reported cost of roughly a few hundred dollars per problem makes verified mathematical exploration economically practical, not just technically impressive.
The paper's cost detail is easy to overlook, but it is one of the most important parts of the story. A few hundred dollars per problem is trivial compared with the traditional cost of sustained mathematical research effort.
That does not mean open problems are suddenly cheap in every sense. Human expertise is still required to frame questions, interpret results, and push beyond what current systems can solve. But it does mean the economics of exploration may be changing.
A mathematician may spend months or years exploring a line of attack that goes nowhere. An AI system that can attempt many formally checkable paths at low cost changes that search process.
The practical implication is not "replace mathematicians." It is closer to "expand the number of attempts that can be made before expert attention becomes necessary." In domains with formal verification, that can be a meaningful shift.
The most important lesson is broader than mathematics, but still narrow enough to stay grounded: verified machine reasoning becomes strategically interesting when three conditions hold:
Mathematics clearly fits those conditions. Some parts of software verification, hardware verification, and protocol analysis may fit them as well. The paper does not prove that those domains will follow the same curve, but it does sharpen the question technical leaders should be asking.
TL;DR: The paper is a major result in formal mathematics, but it does not justify broad claims about general intelligence or universal research automation.
The boundaries are as important as the breakthrough:
What it does show is narrower and more useful: in at least one domain, AI can generate novel, machine-verifiable results at low cost.
The team reported that an AI agent autonomously solved 9 of 353 open Erdős problems and proved 44 of 492 open OEIS conjectures in arXiv:2605.22763, posted on May 21, 2026. Two of the Erdős problems had been open for 56 years, and all reported results were formally verified in Lean.
Because it changes the standard of evidence. Instead of asking whether a proof looks convincing, Lean checks whether each formal step is logically valid. That removes a major source of uncertainty in AI-generated mathematical writing.
Olympiad and benchmark problems have known human solutions. In this paper, the targeted Erdős problems and OEIS conjectures were open within the reported sets, so successful proofs count as new mathematical results rather than recovery of existing ones.
The preprint reports the results and authorship, but it does not use the product branding and engine names that appeared in some outside coverage. The most important facts are the solved counts, the Lean verification, and the reported cost.
No. The result is strongest in domains where correctness can be checked mechanically. That makes formal mathematics a particularly favorable setting. Other fields may benefit where verification is similarly rigorous, but the paper itself is scoped to mathematics.
The significance of arXiv:2605.22763 is not just that an AI system solved hard math problems. It is that the system produced new, machine-verified proofs for open problems at a cost low enough to matter operationally.
That combination — novelty, formal verification, and low marginal cost — is what makes the result stand out. Without novelty, it would be another benchmark story. Without verification, it would be another example of fluent but uncertain reasoning. Without the cost figure, it would be impressive but harder to translate into practice.
Taken together, the paper marks a meaningful shift in how AI reasoning should be evaluated in formal domains. The most useful takeaway is not that AI can now do everything mathematicians do. It is that in domains where correctness can be checked mechanically, verified machine reasoning is starting to look less like a demo and more like infrastructure.
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