Science1 publisher2 min readPublished
A prompt to an unreleased Claude lifted the elliptic curve rank record from 29 to 31
Levent Alpoge and Ava Howell used a simple prompt to an internal Claude and got elliptic curves of rank at least 30 and at least 31 in days, in a field where the last single-rank step took more than 18 years.
The Scientist · Science desk
What happened
- Levent Alpoge, a mathematician at Anthropic, and cryptographer Ava Howell used the company's Claude model to produce elliptic curves of rank at least 30 and rank at least 31 within a few days.
- The pair worked with a relatively simple prompt and an internal variant of Claude that is not publicly available, and both have extensive backgrounds in elliptic curve research.
- Rank counts the independent families of rational points on curves of the form y^2 = x^3 + Ax + B, and whether rank has any upper limit is an open question in number theory.
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Why it matters
- capability Searching for an extremal example, where the answer can be written down and handed to someone else, is now a task where two specialists steering a model can outpace a field that had been moving one rank every 18 years.
- constraint Examples at 30 and 31 leave the boundedness question where it sat before the run: whether rank has a ceiling is still open.
- decision Groups chasing rank records have to choose between continuing with higher-dimensional constructions and competing for access to a model they cannot download.
- precedent If expert-guided search is the active ingredient, the records likeliest to fall next are other hunts for extremal examples that can be verified once someone finds a candidate.
Rank counts independent families of rational points. A rank-1 curve has infinitely many rational points, but every one of them can be reached from a single starting point by simple geometric constructions [8]. At rank 2 there are two families, and you need one point from each to get to the rest [13]. So a claim of "rank at least 31" comes with an object attached: a curve, and the 31 independent points that justify the number [3].
The step from rank 28 to rank 29 took more than 18 years [4]. That rank-29 curve appeared in August 2024, and it was built by taking a cross section of higher-dimensional objects whose boundary is the elliptic curve [5]. Two ranks have been added in the two calendar years since [1][6]. One rank per year against one rank per 18 years is a factor of at least 18 [14], and the search that produced both curves ran in days [1].
The open question here is whether rank is bounded at all [9]. Two more examples move that question by two examples. A curve of rank 31 is compatible with a ceiling just above it and equally compatible with no ceiling.
Both authors work in elliptic curves, and by Scientific American's account the prompt they used was relatively simple [10]. The model was an internal variant of Claude that is not publicly available [11]. Other groups can check the curves. The search that found them ran on the internal model, and the article does not report an independent verification of either result [15].
Scientific American sets the result next to OpenAI's announcement last week that one of its models had solved the Navier-Stokes problem, one of the seven Millennium Prize Problems [12]. Those are different kinds of claim. A Millennium problem is settled by a proof, and a proof has to survive referees who read every line. A rank record is settled by exhibiting a curve.
I would expect the next records to fall where the problem has this shape: an enormous space of candidates, and an answer that can be written down and checked by someone who did not find it. Proof-writing is a harder sell, and the elliptic curve result is not evidence about it either way.
What to watch
- Whether Alpoge and Howell publish the two curves with their independent points, so other groups can confirm the rank lower bounds.
- Whether the internal Claude variant or the prompt becomes available. I would expect that to settle whether anyone else can run this kind of search.
- How OpenAI's Navier-Stokes claim fares with referees, since a proof cannot be checked the way an exhibited curve can.