castedo

I know it's temping to hope there is a single simple factor that does the "real" work, but this feat of mathematics is likely best explained by multiple interacting factors, one of them being the mathematical insights of the human mathematician that tweeted the counterexample. I don't doubt that an LLM is also one of the multiple factors.

It is premature to assume the author is not going to share more information in the future about the mathematical insights to narrow down the search space for this counterexample.

Fable "whipping up" SymPy like a "calculator app" is one such interaction that seems very plausible (in addition to SymPy providing feedback when training models). The scale of compute available to an Anthropic employee for such SymPy calls when using Fable is likely one of multiple factors for this counterexample being found in 2026. Unfortunately, we aren't going to be able to really know the various factors that best explain why an Anthropic employee was able to announce a counterexample this past weekend. For all we know, the counterexample was found by Anthropic employees months ago and used for training the Fable model used this past weekend.

I doubt Anthropic will share the details (or at least the full true details). The mystery of the magic makes for much better marketing.

I think a reasonable assumption is that there is an interaction between an LLM, a https://en.wikipedia.org/wiki/Computer_algebra_system tool, a human prompting with deep math expertise, and lots of compute that explains hitting upon the remarkable cancellation.

... and vibe subscriptions that we can vibe pay with vibecoin.

well then help us out here: can you be specific?

In "Remarks on the disproof of the unit distance conjecture" (https://arxiv.org/abs/2605.20695) I think Melanie Matchett Wood's remark is the most informative: "It is easy to jump to hasty conclusions, but what we can learn about humans, AI, and mathematics from this development is somewhat subtle. I believe if the level and type of human expertise that is represented on this note had been assembled to find a counterexample to this conjecture a month ago, and those people put in similar amounts of time working on it than they did to reading and thinking about ChatGPT’s solution, the mathematicians would have found a counterexample. However, without the claimed proof by ChatGPT, there is no particular reason anyone would have tried to look for a counterexample, assembled a group of experts with the appropriate expertise, or that the experts would have agreed to turn their attention to this problem."

Some readers might find some of the other remarks more appealing or more informative. I encourage folks to read these remarks rather than the OpenAI marketing video and spin.

To me it sounds a lot like goalpost moving. You're telling me that in 2020 if I showed you a system that can solve an Erdos problem or disprove a conjecture (just recently showed up) you wouldn't be blown away?

I'm not sure what goalpost you are talking about. Regarding 2020: it depends on the framing, how much I know about the conjecture, the details of the computer system. I an easily imagine not being blown away. But I don't really see the relevance of our emotional reactions to computers doing new things we've never seen computers do before. If the goalpost is being "blown away" by what computers can do, then that happened I think around 1990 when I heard a computer program generate a vaguely human sounding voice. In math, I think it happened when I saw Mathematica simplified a huge nasty complicated algebra expression around 2000. I've been "blown away" by new things computers can do many times over the past 36 years.

Woah woah, that's not the conclusion I'm jumping to. ... ok but now we're solving real math problems.

Sounds like we are in agreement then that (1) LLMs can not solve any given Erdos problem and (2) computers are solving more real math problems than they were before.

I honestly do think what the OpenAI group did with an LLM recently is a new milestone worthy of attention if one is interested in computer assisted mathematics. I don't mean to diminish the LLM feat. I just mean to throw shade on the corporate marketing, language, slick video, and spin.

The fact that these "stochastic code extruders" can solve Erdos problems is sort of the proof in the pudding.

This claim is very misleading and not really true. It reflects the kind of exaggeration and spin made by corporate marketing. I would not call this a fact at all. Like many claims made by for-profit marketing, if one looks into the details and think critically about what is being claimed, one can see that consumers are jumping to false conclusions.

That said, it is very cool how an LLM helped human mathematicians in the recent specific Erdos problem solution announced by OpenAI. Just don't jump to the conclusion that anybody can input any Erdos problem into an LLM and a solution will come out the other end.

Similar use cases of useful AI one-shot projects are: demos, proof-of-concept, prototype, exploratory tests, etc... I've AI-generated an HTML/javascript craplet to test how browser security would behave for certain JavaScript API calls. I just wanted to confirm how browser security would behave before I started to spend time hand-coding the non-AI quality software that I wanted to release.

And I say this as somebody who generally has a low opinion of AI generated code and rarely uses it. But it has its place.

Me too, ... more or less. I'm mostly still typing, sometimes copy-and-pasting with typed changes, and rarely copy-and-pasting verbatim. With the caveat that in some cases, like prototypes, proofs-of-concept, and porting code between languages; then maybe many lines are copy-and-pasted verbatim.