The theorem is implied by an older result of Erdos, but is not a result of Erdos. Apparently this is because the connection is something called "Roger's Theorem" that was quite obscure.
"This theorem is somewhat obscure: its only appearance in print is in pages 242-244 of this 1966 text of Halberstam and Roth, where the authors write in a footnote that the result is “unpublished; communicated to the authors by Professor Rogers”. I have only been able to find it cited in three places in the literature: in this 1996 paper of Lewis, in this 2007 paper of Filaseta, Ford, Konyagin, Pomerance, and Yu (where they credit Tenenbaum for bringing the reference to their attention), and is also briefly mentioned in this 2008 paper of Ford. As far as I can tell, the result is not available online, which could explain why it is rarely cited (and also not known to AI tools). This became relevant recently with regards to Erdös problem 281, posed by Erdös and Graham in 1980, which was solved recently by Neel Somani through an AI query by an elegant ergodic theory argument. However, shortly after this solution was located, it was discovered by KoishiChan that Rogers’ theorem reduced this problem immediately to a very old result of Davenport and Erdös from 1936. Apparently, Rogers’ theorem was so obscure that even Erdös was unaware of it when posing the problem!"
So it would be able to produce the training data but with sufficient changes or added magic dust to be able to claim it as one's own.
Legally I think it works, but evidence in a court works differently than in science. It's the same word but don't let that confuse you and don't mix them both.
The model doesn't know what its training data is, nor does it know what sequences of tokens appeared verbatim in there, so this kind of thing doesn't work.
It's not the searching that's infeasible. Efficient algorithms for massive scale full text search are available.
The infeasibility is searching for the (unknown) set of translations that the LLM would put that data through. Even if you posit only basic symbolic LUT mappings in the weights (it's not), there's no good way to enumerate them anyway. The model might as well be a learned hash function that maintains semantic identity while utterly eradicating literal symbolic equivalence.
I saw weird results with Gemini 2.5 Pro when I asked it to provide concrete source code examples matching certain criteria, and to quote the source code it found verbatim. It said it in its response quoted the sources verbatim, but that wasn't true at all—they had been rewritten, still in the style of the project it was quoting from, but otherwise quite different, and without a match in the Git history.
It looked a bit like someone at Google subscribed to a legal theory under which you can avoid copyright infringement if you take a derivative work and apply a mechanical obfuscation to it.
People seem to have this belief, or perhaps just general intuition, that LLMs are a google search on a training set with a fancy language engine on the front end. That's not what they are. The models (almost) self avoid copyright, because they never copy anything in the first place, hence why the model is a dense web of weight connections rather than an orderly bookshelf of copied training data.
Picture yourself contorting your hands under a spotlight to generate a shadow in the shape of a bird. The bird is not in your fingers, despite the shadow of the bird, and the shadow of your hand, looking very similar. Furthermore, your hand-shadow has no idea what a bird is.
Threatening violence*, even in this virtual way and encased in quotation marks, is not allowed here.
Edit: you've been breaking the site guidelines badly in other threads as well. (To pick one example of many: https://news.ycombinator.com/item?id=46601932.) We've asked you many times not to.
I don't want to ban your account because your good contributions are good and I do believe you're well-intentioned. But really, can you please take the intended spirit of this site more to heart and fix this? Because at some point the damage caused by poisonous comments is worse.
* it would be more accurate to say "using violent language as a trope in an argument" - I don't believe in taking comments like this literally, as if they're really threatening violence. Nonetheless you can't post this way to HN.
I don't think it is dispositive, just that it likely didn't copy the proof we know was in the training set.
A) It is still possible a proof from someone else with a similar method was in the training set.
B) something similar to erdos's proof was in the training set for a different problem and had a similar alternate solution to chatgpt, and was also in the training set, which would be more impressive than A)
It is still possible a proof from someone else with a similar method was in the training set.
A proof that Terence Tao and his colleagues have never heard of? If he says the LLM solved the problem with a novel approach, different from what the existing literature describes, I'm certainly not able to argue with him.
Does it matter if it copied or not? How the hell would one even define if it is a copy or original at this point?
At this point the only conclusion here is:
The original proof was on the training set.
The author and Terence did not care enough to find the publication by erdos himself
https://terrytao.wordpress.com/2026/01/19/rogers-theorem-on-...
"This theorem is somewhat obscure: its only appearance in print is in pages 242-244 of this 1966 text of Halberstam and Roth, where the authors write in a footnote that the result is “unpublished; communicated to the authors by Professor Rogers”. I have only been able to find it cited in three places in the literature: in this 1996 paper of Lewis, in this 2007 paper of Filaseta, Ford, Konyagin, Pomerance, and Yu (where they credit Tenenbaum for bringing the reference to their attention), and is also briefly mentioned in this 2008 paper of Ford. As far as I can tell, the result is not available online, which could explain why it is rarely cited (and also not known to AI tools). This became relevant recently with regards to Erdös problem 281, posed by Erdös and Graham in 1980, which was solved recently by Neel Somani through an AI query by an elegant ergodic theory argument. However, shortly after this solution was located, it was discovered by KoishiChan that Rogers’ theorem reduced this problem immediately to a very old result of Davenport and Erdös from 1936. Apparently, Rogers’ theorem was so obscure that even Erdös was unaware of it when posing the problem!"