https://twitter.com/vladtenev/status/1994922827208663383
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.
GPT-5 solved Erdős problem #848 (combinatorial number theory):
https://cdn.openai.com/pdf/4a25f921-e4e0-479a-9b38-5367b47e8...
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...
He was cited https://the-decoder.com/leading-openai-researcher-announced-...
Where a while back OpenAI made a misleading claim about solving some of these problems.
"Don't be curmudgeonly. Thoughtful criticism is fine, but please don't be rigidly or generically negative."
> By that logic, we've had LLMs since the 60s!
From a bit earlier[0], actually:
Progressing to the 1950s and 60s
We saw the development of the first language models.
0 - https://ai-researchstudies.com/history-of-large-language-mod...
> 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
roughly, what lean theorem (and statement on the website) asks is this: take some numbers t_i, for each of them form all the powers t_i^j, then combine all into multiset T. Barring some necessary conditions, prove that you can take subset of T to sum to any number you want
what Erdosh problem in the paper asks is to add one more step - arbitrarily cut off beginnings of t_i^j power sequences before merging. Erdosh-and-co conjectured that only finite amount of subset sums stop being possible
"subsets sum to any number" is an easy condition to check (that's why "olympiad level" gets mentioned in the discussion) - and it's the "arbitrarily cut off" that's the part that og question is all about, while "only finite amount disappear" is hard to grasp formulaically
so... overhyped yes, not actually erdos problem proven yes, usual math olympiad level problems are solvable by current level of Ai as was shown by this year's IMO - also yes (just don't get caught by https://en.wikipedia.org/wiki/AI_effect on the backlash since olympiads are haaard! really!)
Everything is amazing and nobody is happy: https://www.youtube.com/watch?v=nUBtKNzoKZ4
"How quickly the world owes him something, he knew existed only 10 seconds ago"
1. https://www.claymath.org/millennium-problems/
1.