A "Failed" 54-Hour Run That Rewrote One Line in the Number Theory Textbooks

Let's state the conclusion up front: Claude did not prove the Riemann hypothesis. The problem, open since 1859 and carrying a million-dollar Clay Institute bounty, remains unsolved. But on August 10, Anthropic published a result that made number theorists sit up — an unreleased research version of Claude, running inside the Claude Code agentic harness for roughly a day and a half, pushed the proven share of the Riemann zeta function's zeros lying on the critical line from 41.6% to 67.2% (Forbes report).

What does that number mean? In the 37 years prior, the entire cumulative progress by human mathematicians on this record was 0.8 percentage points. Claude's single run moved it by 25.6 points — the largest single advance in the problem's 165-year history. Deedy Das, a partner at Menlo Ventures, called it the biggest result in analytic number theory since Yitang Zhang's bounded prime gaps in 2013.

How It Did It: 31M Tokens, 60 Subagents, 650 Dead Ends

The engineering details of this run are a specimen worth dissecting on their own (figures below are from Forbes' account of the Anthropic paper):

  • Two sessions produced 31 million output tokens, deployed 60 subagents, and ran 2,400 shell commands;
  • The first session generated 650 candidate ideas — every one a dead end;
  • The session that produced the proof ran for roughly a day and a half, continuously checking its claims numerically against thousands of known zeta zeros.

Session one: 650 consecutive failures. Session two: record broken. That is what a research program looks like when failure is priced in tokens.

Methodologically, Claude performed a textbook case of cross-school synthesis: one line is a series of papers by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh adapting Montgomery's classic technique so it no longer assumes the Riemann hypothesis; the other is a 2000 paper by Bombieri. Both lines had been sitting in the literature for years — nobody had ever "read them together." Number theory is specialized to the point where an entire career fits inside one subfield, and Claude's advantage is precisely that it has no specialization — it read all of the literature at once.

The identity of the guide is even more telling: per the Wall Street Journal, Jarred Sumner — whose formal mathematical education ended after one semester of high-school geometry — steered the whole process by repeatedly encouraging the model to keep trying, and even misspelled "Riemann" in his opening prompt. That is not an anecdote; that is a frontal challenge to the monopoly of credentialed expertise.

Verification: Machine-Checked, Then Human-Reviewed by Giants

Anthropic did not ask anyone to take its word for it. Two of its research mathematicians, Levent Alpöge and Ralph Furman, worked through the argument. Claude then produced a formalization in Lean that passes machine verification — a level of scrutiny most published mathematics never receives. The external reviewers carry real weight too: Brian Conrey, whose 1989 proof anchored the modern record, and Dan Goldston, co-author of part of the work Claude built on.

The honest limits are stated plainly: the result has not been through conventional peer review; the model that produced it is unreleased, so the run cannot be reproduced end to end; and Anthropic itself says it does not expect this line of attack to lead to a full proof of the Riemann hypothesis.

So What: Expensive Knowledge, Cheap Synthesis

One line from the Forbes analysis deserves to be written down by anyone who does research: expensive knowledge, cheap synthesis. Decades of specialist literature are expensive; the compute bill for connecting it all is trivial by comparison. That ratio is the business case for frontier models in research — until this week it was a pitch-deck claim, and now it is a machine-verified theorem.

The takeaway for a general reader is perhaps plainer: the next time you face a problem that "only experts are qualified to touch," remember this picture — a person who never finished high-school geometry, one prompt with a misspelled name, and a model willing to fail 650 times, together touched the edge of number theory's holy grail. Professional barriers never protected the truth; they protected admission.