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Hacker News Hacker News> This is a nice solution, and impressive to be found by AI, although the proof is (in hindsight) very simple, and the surprising thing is that Erdos missed it. But there is definitely precedent for Erdos missing easy solutions!
> Also this is not the problem as posed in that paper
> That paper asks a harder version of this problem. The problem which has been solved was asked by Erdos in a couple of later papers.
> One also needs to be careful about saying things like 'open for 30 years'. This does not mean it has resisted 30 years of efforts to solve it! Many Erdos problems (including this one) have just been forgotten about it, and nobody has seriously tried to solve it.[1]
And, indeed, Boris Alexeev (who ran the problem) agrees:
> My summary is that Aristotle solved "a" version of this problem (indeed, with an olympiad-style proof), but not "the" version.
> I agree that the [BEGL96] problem is still open (for now!), and your plan to keep this problem open by changing the statement is reasonable. Alternatively, one could add another problem and link them. I have no preference.[2]
Not to rain on the parade out of spite, it's just that this is neat, but not like, unusually neat compared to the last few months.
[1] https://twitter.com/thomasfbloom/status/1995083348201586965
[2] https://www.erdosproblems.com/forum/thread/124#post-1899
A reality where we are talking about if the problem solved by the automated model using formal verification was too easy.
Don't get me wrong, I am not saying any of this means we get AGI or something or even if we continue to see improvements. We can still appreciate things. It doesn't need to be a binary. What a time to be alive!
These people treat math research as if it is a math homework assignment. There needs to be an honest discussion about what the LLM is doing here. When you bang your head against a math problem you blindly try a bunch of dumb ideas that don't actually work. It wastes a lot of paper. The LLM automates a lot of that.
It is actually pretty cool that modern AI can help speed this up and waste less paper. It is very similar to how classical AI (Symbolica) sped up math research and wasted less paper. But we need to have an honest discussion about how we are using the computer as a tool. Instead malicious idiots like Vlad Tenev are making confident predictions about mathematical superintelligence. So depressing.
GPT-5 solved Erdős problem #848 (combinatorial number theory):
https://cdn.openai.com/pdf/4a25f921-e4e0-479a-9b38-5367b47e8...
https://arxiv.org/html/2510.19804v1#Thmtheorem3
> In over a dozen papers, beginning in 1976 and spanning two decades, Paul Erdős repeatedly posed one of his “favourite” conjectures: every finite Sidon set can be extended to a finite perfect difference set. We establish that {1, 2, 4, 8, 13} is a counterexample to this conjecture.
related article:
> Is Math the Path to Chatbots That Don’t Make Stuff Up?
https://www.nytimes.com/2024/09/23/technology/ai-chatbots-ch...
1. https://www.claymath.org/millennium-problems/
1.
On one hand, as the nice sketch provided below by tsaf confirms, the final proof is quite simple and elementary - indeed, if one was given this problem in a maths competition (so therefore expected a short simple solution existed) I'd guess that something like the below would be produced. On the other hand, if something like this worked, then surely the combined talents of Burr, Erdős, Graham, and Li would have spotted it.
Normally, this would make me suspicious of this short proof, in that there is overlooked subtlety. But (a) I can't see any and (b) the proof has been formalised in Lean, so clearly it just works!
Perhaps this shows what the real issue in the [BEGL96] conjecture is - namely the removal of 1 and the addition of the necessary gcd condition. (And perhaps at least some subset of the authors were aware of this argument for the easier version allowing 1, but this was overlooked later by Erdős in [Er97] and [Er97e], although if they were aware then one would hope they'd have included this in the paper as a remark.)
At the moment I'm minded to keep this as open, and add the gcd condition in the main statement, and note in the remarks that the easier (?) version allowing 1 and omitting the gcd condition, which was also asked independently by Erdős, has been solved."
The commentator is saying: "I can't believe this famous problem was solved so easily. I would have thought it was a fake proof, but the computer verified it. It turns out the solution works because it addresses a slightly different set of constraints (regarding the number 1) than what Erdős originally struggled with. (Generated by Gemini)
Software engineering is more subtle since you actually care about the performance and semantics of a program. Unless you get really good at program specification, it would be difficult to fully automate.
But with math the only thing you care about is whether the moves you made were right.
Sure it suffers from amnesia and cannot get basic things right sometimes, but one is a design limitation that can be overcome and the other a possible problem caused by training (we’re discovering that overt focus during training on adherence to prompt somewhat lobotomizes their general competence).