An Anthropic employee with no serious training in mathematics told it to have a go, then walked away for a day and a half. It found no proof, but it lifted the share of the Riemann zeta function's zeros known to obey the hypothesis from 41.6% to 67.2% — a bound decades of human work had inched along.
The whole instruction was to make a genuine attempt at proving the Riemann hypothesis. The person who gave it, a member of Anthropic's staff with no serious training in mathematics, then left an unreleased Claude model to work the problem alone and came back a day and a half later.
What it returned is not a solution, and the $1 million offered for a general proof is still unclaimed. The hypothesis, open for more than 150 years, holds that every non-trivial zero of the zeta function lies on one particular line; settle it and you settle how the prime numbers are distributed. Nobody has proved that for all of them. What mathematicians have done instead is prove it for a fraction of them and push that fraction up by degrees, and decades of work had carried it to 41.6%. The model took it to 67.2% — two-thirds of infinitely many zeros where there had been two-fifths.
The first attempt went nowhere. The model generated and tried 650 ideas, and not one of them worked. What followed was the day and a half: it coordinated about 60 subagents through 2,400 shell commands and hundreds of Python scripts, spending 31 million tokens of output. A footnote in the paper accounts for all 60 — two produced the central mathematical insights, 13 fed that pair supporting ideas, 30 tried and came up with nothing new, 13 acted as checkers verifying the arguments were sound, and the last two helped draft the first version of the paper.
Two of Anthropic's own mathematicians, Levent Alpöge and Ralph Furman, went through the work to understand what the model had done and how it stood against the prior literature. The result was also written out in Lean, the open-source proof assistant, where it passes the standard validation tool.
Results of roughly this shape have been arriving all year. OpenAI recently published ten significant mathematical results proved by Astra, a model it keeps in house, and a separate effort at Anthropic showed the long-standing Jacobian conjecture to be false.
Who gets the credit for work done this way is contested. A group of leading mathematicians signed a public statement in June warning that AI could erode something the discipline holds dear — the expectation that a proof can be traced back to named people who claim credit for what they found and answer for whether it is right.
Timothy Gowers, a Fields Medal recipient, replied on his blog by asking whether AI's expanding role might change mathematics in a way more intricate, and more beneficial, than the declaration suggested. If mathematics reached a stage where theorems were no longer tied to particular mathematicians, he wrote, that need be no more troubling than stars not carrying the names of the astronomers who spotted them — most stars carry no name at all.