Comment by Legend2440
1 day ago
Just this week an LLM found a counterexample to math problem that's been widely studied for over a century: https://en.wikipedia.org/wiki/Jacobian_conjecture
>The conjecture was first stated for two variables by Ludwig Kraus in 1884[1] and then stated in full generality in 1939 by Ott-Heinrich Keller.[2] It was subsequently widely publicized by Shreeram Abhyankar,[3] as an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus.
>The Jacobian conjecture was notorious for the large number of published and unpublished proofs that turned out to contain subtle errors.[4][5]
>On July 19, 2026, Anthropic employee and mathematician Levent Alpöge presented an explicit counterexample in three-dimensional space, discovered by Anthropic's large language model Claude Fable 5, which disproves the conjecture for n > 2
If that won't convince you that LLMs do more than parrot existing ideas, you've got your head in the sand.
Is there any evidence that the discovery was made by Claude and not by Levent Alpöge himself? The only sources listed in the wikipedia article are an X post and a news article that references the post.
You think Alpoge just had the solution for the Jacobian Conjecture in his back pocket, just sort of waiting to deploy at his next gig?
He could have been paid by Anthropic to run a brute-force computer search. The counterexample looks short enough that, given a portion of Anthropic’s computing power and a sufficiently smart algorithm, it could be found by brute force.
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This is a very well studied problem. Well-known mathematicians have spent a lot of effort on it - Yitang Zhang wrote his entire PhD thesis on it back in 1991.
It is deeply unlikely that a random guy at Anthropic just happened to solve it so they could pass it off as the LLM's work.
> If that won't convince you that LLMs do more than parrot existing ideas, you've got your head in the sand.
It doesn't.
In a nearby comment: https://news.ycombinator.com/item?id=48983413
> In mathematics (including information science, CS) there are all sorts of problems that are essentially searches for a solution, and many have the property that the search is computationally difficult, but verifying the solution is relatively cheap. E.g. finding integers such that a^2 + b^2 = c^2 isn't easy, but given a claim that some proposed <a, b, c> satisfies this equation is easy to check. The LLM is like that: it solves a search problem that can be fairly hard.
Funnily enough, one of the attempted solves in the litterature is exploring the problem space in two-dimensional space; the LLM found one in three-dimensional space.
So far we know very little as to why and how it found the solution.
It may very well have been directed to brute force 3D space, or even "elected" to" by expanding the known-failed 2D approach to 3D as pure mimicry.
> It does so unreliably, but if you can cheaply verify the solution, there is a win there.
This circles back to what the LLM advocates are pushing for: build the harness that keeps the agent in check, guardrails all the way because it's driving like a demolition derby.
I think when we're at the point of No True Intellectual Achievementing Smale's Mathematical Problems for the Next Century, everyone's premises have drifted too far apart for discussion to be reasonable.
You do indeed have your head in the sand, that's for sure.
You are desperately searching for ways to excuse it, to explain how it can't be what it obviously is.
On this matter I'm not searching for excuses, I am reserving my judgement; all we have is a tweet with the counterexample. We don't know how the counterexample was built nor found. It's just not useful to speculate.