OpenAI · AIGC · AI Programming

Claude Ran a 31M-Token Math Sprint and Broke a 6-Year Record on the Riemann Hypothesis

By 爱吃的小肥羊 · Aug 11, 2026

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A frontier model, given a hard open problem and left to run, produced a mathematically meaningful result that four domain experts verified and that survives formal proof checking. The jump from 41.6% to 67.2% doesn't solve the Riemann Hypothesis, but it shows AI can now advance a stalled mathematical record when the right prior literature exists to build on.

Summary

Anthropic let a research Claude loose on the Riemann Hypothesis with minimal human steering. After 650 dead-end ideas in the first attempt, a second run coordinated roughly 60 sub-agents across a day and a half, burning 31 million output tokens. The model didn't prove the hypothesis, but it raised a related lower bound by over 25 percentage points — the first movement on that metric in six years. The result was verified by four mathematicians, including Brian Conrey, who set the prior two-fifths record in 1989, and the proof was formalized in Lean with no `sorry` placeholders. The human in the loop, Jarred Sumner, is a 16-year-old dropout with no math background; his contributions were limited to messages like "keep going" and "believe in yourself."

Takeaways

— Claude tested roughly 650 ideas in its first attempt; all failed.

— A second attempt used about 60 sub-agents inside Claude Code to split the work into ideation, computation, counterexample search, proof review, and paper writing.

— Only 2 of the 60 sub-agents produced key mathematical ideas; roughly half failed outright.

— The two sessions consumed 31 million output tokens and ran about 2,400 shell commands.

— The proven proportion of non-trivial zeros on the critical line jumped from ~41.67% (set in 2020) to 67.25%.

— Four mathematicians — Levent Alpöge, Ralph Furman, Brian Conrey, and Dan Goldston — reviewed and verified the result.

— The proof was formalized in Lean, passed the comparator tool, and contains no `sorry` placeholders.

— Anthropic stated the techniques are not expected to scale directly to a full proof of the Riemann Hypothesis.

— The human operator, Jarred Sumner, is not a mathematician and contributed only encouragement, not mathematical direction.

Conclusions

The result's credibility rests less on the AI's output and more on the verification pipeline: four human experts plus a Lean formal proof that passes comparator checks. That's a higher bar than most AI math demos clear.

Half the sub-agents failed, and only two produced key ideas — a reminder that scaling agent count doesn't linearly produce insight. The architecture succeeded by generating enough shots that a few stuck.

The model's dependence on recent human papers (Baluyot, Goldston, Suriajaya, Turnage-Butterbaugh, Bombieri) suggests the breakthrough was combinatorial assembly of existing results rather than novel theory generation. That's still useful, but it bounds the kind of problem this approach can currently crack.

Anthropic's public release of the paper, process log, and formal proof sets a transparency standard that makes independent verification possible — a contrast to the typical AI demo that ships a cherry-picked output with no audit trail.

Concepts & terms

Riemann Hypothesis

An 1859 conjecture that all non-trivial zeros of the Riemann zeta function lie on the critical line where the real part equals 1/2. It is one of the seven Clay Millennium Prize Problems and remains unproven after 167 years.

Non-trivial zeros

The zeros of the Riemann zeta function that are not negative even integers. Their distribution is deeply connected to the distribution of prime numbers, and proving they all lie on the critical line is the core of the Riemann Hypothesis.

Lower bound on the critical line

Since proving all non-trivial zeros lie on the critical line remains out of reach, mathematicians instead prove what proportion must lie there. This proportion has been raised incrementally over decades — from Hardy's infinite-but-unspecified in 1914 to 41.6% in 2020, and now 67.2% by Claude.

Lean formal proof

Lean is a proof assistant and programming language that allows mathematical proofs to be written as machine-checkable code. A proof without `sorry` placeholders means every logical step has been fully verified by the Lean kernel.

Comparator (Lean verification tool)

A tool that checks whether a submitted Lean proof actually proves the stated theorem, preventing the common trick of proving a weaker or different statement and claiming the original result.

From the discussion

Featured comments

JohnYan

Is this because they feel the company gives them too many tokens and they want to hit a KPI?

爱吃的小肥羊

Their quota is probably unlimited, after all, 80% profit [crying]

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