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The Claude logo is displayed on a computer screen photographed through a magnifying glass in Creteil, France, on April 21, 2026. The NSA is reported to have used Claude Mythos Preview despite a ban on Anthropic for United States government agencies. (Photo by Samuel Boivin\/NurPhoto via Getty Images)<\/p>\n
NurPhoto via Getty ImagesAnthropic gave the problem to an unreleased research version of Claude running inside Claude Code.<\/p>\n
Number theorists spent 37 years moving one number by less than a single percentage point. The number is the proven share of the Riemann zeta function’s zeros<\/a> that sit on the critical line, the place where one of the most famous unsolved problems in mathematics says all of them belong. Decades of human refinement had carried it to 41.6%. On August 10, Anthropic published a result<\/a> from an unreleased research version of Claude that moved it to 67.2% in roughly a day and a half.<\/p>\n The Riemann hypothesis has been open since 1859. It predicts the hidden structure behind how prime numbers are distributed, and it carries a million-dollar Clay Institute bounty. The hypothesis itself remains open. What Claude established is that at least two-thirds of the zeros behave the way it predicts, the largest single advance in the record’s history.<\/p>\n The investing story is the mechanism: Claude built the proof from mathematics humans had already published, and the record moved when a single reader could hold the entire literature at once.<\/p>\n What Claude Actually Did<\/p>\n Anthropic gave the problem to a research version of Claude running inside Claude Code, its agentic coding harness, and let it work. The scale of the run<\/a> is its own story:<\/p>\n Two sessions produced 31 million output tokensThe model deployed 60 subagents and ran 2,400 shell commandsThe first session generated 650 candidate ideas, every one a dead endThe session that produced the proof ran for roughly a day and a half, checking its claims numerically against thousands of known zeta zeros<\/p>\n The first session failed 650 times. The second one broke the record. That is what a research program looks like when failure is priced in tokens.<\/p>\n The proof combines two lines of existing work. One is a series of papers by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh, which adapted a classic technique of Montgomery’s so it no longer needs to assume the Riemann hypothesis. The other is a 2000 paper by Bombieri. The ingredients were in the literature. Claude’s contribution was treating together what earlier work had handled in separate pieces.<\/p>\n How The Proof Was Checked<\/p>\n Anthropic did not ask anyone to take its word. Two of its research mathematicians, Levent Alp\u00f6ge and Ralph Furman, worked through the argument. Claude produced a formalization in Lean, the proof-assistant language, and the formalization passes machine verification. A machine-checked formal proof is a level of scrutiny most published mathematics never receives.<\/p>\n Two outside number theorists examined the paper as well, and the choice of reviewers says something on its own. One is Brian Conrey, whose 1989 proof anchored the modern record. The other is Dan Goldston, who co-wrote part of the work Claude built on.<\/p>\n The honest limits are in the post too. The result has not been through conventional peer review. The run cannot be reproduced end to end, because the model that produced it is unreleased. And Anthropic states plainly that it does not expect Claude’s techniques to lead to a proof of the Riemann hypothesis itself.<\/p>\n Where The Advance Came From<\/p>\n The tempting story is that an AI out-thought the mathematicians. The evidence points somewhere more specific. Number theory is organized into specializations deep enough that a career fits inside one of them, and the researchers behind the two lines Claude combined publish in the same journals without needing each other’s machinery. Claude read both lines at once and found that they fit. The advance was sitting in the published literature, waiting for a reader with no specialization.<\/p>\n Deedy Das<\/a>, a partner at the venture firm Menlo Ventures, put the scale of the move in context:<\/p>\n This insane result from Claude is the biggest in analytic number theory since bounded prime gaps in 2013. It boosted the proven fraction of Riemann zeta zeros on the critical line by 25.6pts. In the 37yrs prior, mathematicians moved it 0.8pts.<\/p>\n The 2013 reference is Yitang Zhang’s celebrated proof on gaps between primes, the last time a single result moved this field that hard.<\/p>\n Research As A Compute Bill<\/p>\n The run consumed 31 million output tokens. The number that matters sits next to it: the proof leaned on decades of published specialist work, and the compute bill for connecting all of it was trivial by comparison. Expensive knowledge, cheap synthesis. That ratio is the business case for frontier models in research, and until this week it was a pitch-deck claim rather than a theorem.<\/p>\n The timing gives the demonstration an audience. Anthropic’s bankers began meeting institutional investors<\/a> in July ahead of a Nasdaq listing that Bloomberg reports<\/a> could come as soon as October, at a $965 billion valuation. The doubt those meetings keep circling, per the Wall Street Journal<\/a>, is whether frontier models are converging into interchangeable products. A machine-verified advance in analytic number theory is the sharpest available counterevidence, a capability gap you can state as a theorem.<\/p>\n Anthropic has not claimed the method generalizes, and nobody knows yet whether it does. The economics do not need generality. If even a narrow class of synthesis problems yields to token spend, every serious lab will spend tokens to find out which problems those are. Runs like this land as demand on the infrastructure underneath all of them, whichever model ends up ahead. The models are learning to read everything humans have already written, and selling them the capacity to do it is the business that gets paid either way.<\/p>\n","protected":false},"excerpt":{"rendered":"The Claude logo is displayed on a computer screen photographed through a magnifying glass in Creteil, France, on…\n","protected":false},"author":2,"featured_media":138796,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[1276,53,3154,182,37952,2213,3261,38107,28681,64140],"class_list":["post-138795","post","type-post","status-publish","format-standard","has-post-thumbnail","category-anthropic","tag-ai-models","tag-anthropic","tag-anthropic-claude","tag-claude","tag-claude-fable","tag-claude-mythos","tag-dario-amodei","tag-fable-5","tag-llm-benchmarks","tag-llm-leaderboard"],"_links":{"self":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts\/138795","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/comments?post=138795"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts\/138795\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/media\/138796"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/media?parent=138795"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/categories?post=138795"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/tags?post=138795"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}