Anthropic said on Monday that a model it has not yet shipped made measurable headway on the Riemann hypothesis, raising the lower bound below which the conjecture is known to hold. The problem has resisted proof since the middle of the nineteenth century, and a general solution still carries a $1 million bounty that nobody has collected. That prize remains unclaimed, and this work does not claim it. What the result does do is reopen an argument the field has been having all year about whether these systems can originate mathematical ideas or only rearrange existing ones.
The method is the part likely to draw the most attention. An Anthropic staff member with no serious mathematical training asked the model to attempt a proof and then stepped back, letting it run and organise its own work for roughly a day and a half. Over that period it worked through 650 separate candidate approaches, distributed the labour across 60 subordinate agents, and consumed 31 million tokens getting there.
The division of that labour was strikingly lopsided, and the paper's own footnotes lay it out. Two of the 60 agents produced the mathematical ideas that carried the result. Thirteen fed material to those two. Thirty tried to generate something usable and came up with nothing. A further thirteen acted as checkers, testing whether the arguments held. The last two drafted the paper.
Verification did not stop at the model's own checkers. Two mathematicians on Anthropic's staff confirmed the finding, and the argument was then formalised in Lean, the open-source proof assistant. Formalisation matters here more than it might elsewhere: it converts a claim that reads convincingly into one a machine has checked line by line, which is precisely the assurance a result produced this way needs.
None of this arrives in isolation. Language models have been chipping away at a run of Erdős problems through the year, and each jump in model capability has produced correspondingly better output. OpenAI recently published a set of ten significant results credited to an internal system it calls Astra. A separate Anthropic effort disproved the Jacobian conjecture, a question that had stood for decades.
The accumulating results have landed in the mathematical community as both an opportunity and a problem. A declaration published in June, signed by a number of well-known figures in the discipline, warned that mathematics risks losing something it depends on — the principle that a proof belongs to named people who take credit for it and answer for whether it is correct. Strip that out, the argument runs, and the machinery of accountability that keeps the subject honest goes with it.
The field has not settled on a response. Timothy Gowers, a Fields medallist, replied to that declaration in a blog post taking the opposite view, or at least a more relaxed one. If mathematics ends up somewhere that theorems are no longer tied to individual mathematicians, he suggested, the outcome might prove no more troubling than the fact that stars are not named for the astronomers who found them, and that most of them carry no name at all.
