Interesting that in Terrance Tao's words: "though the new proof is still rather different from the literature proof)"
And even odder that the proof was by Erdos himself and yet he listed it as an open problem!
And even odder that the proof was by Erdos himself and yet he listed it as an open problem!
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is in pages 242-244 of this 1966 text of Halberstam and Roth, where the authors write in a footnote that the result is “unpublished; communicated to the authors by Professor Rogers”.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!"
It really contextualizes the old wisdom of Pythagoras that everything can be represented as numbers / math is the ultimate truth
They create concepts in latent space which is basically compression which forces this
But I'm not a mathematics expert if this is the real official definition I'm fine with it. But are you though?
consider estimating the position of an object from noisy readings. One presumes that position to exist in some sense, and then one can estimate it by combining multiple measurements, increasing positioning resolution.
its any variable that is postulated or known to exist, and for which you run some fitting procedure
It doesn't matter if ai is in a hype cycle or not it doesn't change how a technology works.
Check out the yt videos from 1blue3brown he explains LLMs quite well. .your first step is the word embedding this vector space represents the relationship between words. Father - grandfather. The vector which makes a father a grandfather is the same vector as mother to grandmother.
You the use these word vectors in the attention layer to create a n dimensional space aka latent space which basically reflects a 'world' the LLM walks through. This makes the 'magic' of LLMs.
Basically a form of compression by having higher dimensions reflecting kind a meaning.
Your brain does the same thing. It can't store pixels so when you go back to some childhood environment like your old room, you remember it in some efficient (brain efficient) way. Like the 'feeling' of it.
That's also the reason why an LLM is not just some statistical parrot.
It does change what people say about it. Our words are not reality itself; the map is not the territory.
Are you saying people should take everything said about LLMs at face value?
It's the reason why I'm here because we discuss more technically about technology
I spend too much time here and decided to delete my account to interact less.
It's partially working though
I know that at least some LLM products explicitly check output for similarity to training data to prevent direct reproduction.
It's great business to minimally modify valuable stuff and then take credit for it. As was explained to me by bar-certified counsel "if you take a recipe and add, remove or change just one thing, it's now your recipe"
The new trend in this is asking Claude Code to create a software on some type, like a Browser or a DICOM viewer, and then publishing that it's managed to do this very expensive thing (but if you check source code, which is never published, it probably imports a lot of open source dependencies that actually do the thing)
Now this is especially useful in business, but it seems that some people are repurposing this for proving math theorems. The Terence Tao effort which later checks for previous material is great! But the fact that the Section 2 (for such cases) is filled to the brim, and section 1 is mostly documented failed attempts (except for 1 proof, congratulations to the authors), mostly confirms my hypothesis, claiming that the model has guards that prevent it is a deus ex machina cope against the evidence.
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 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.
Carbon copy would mean over fitting
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.
But honestly source = "a knuckle sandwich" would be appropriate 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.
https://news.ycombinator.com/showhn.html
* 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.
this is a verbatim quote from gemini 3 pro from a chat couple of days ago:
"Because I have done this exact project on a hot water tank, I can tell you exactly [...]"
I somehow doubt it an LLM did that exact project, what with not having any abilities to do plumbing in real life...
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)
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
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.
Tao et al. didn't know of the literature proof that started this subthread.
> He speculated that "the formulation [of the problem] has been altered in some way"....
[snip]
> More broadly, I think what has happened is that Rogers' nice result (which, incidentally, can also be proven using the method of compressions) simply has not had the dissemination it deserves. (I for one was unaware of it until KoishiChan unearthed it.) The result appears only in the Halberstam-Roth book, without any separate published reference, and is only cited a handful of times in the literature. (Amusingly, the main purpose of Rogers' theorem in that book is to simplify the proof of another theorem of Erdos.) Filaseta, Ford, Konyagin, Pomerance, and Yu - all highly regarded experts in the field - were unaware of this result when writing their celebrated 2007 solution to #2, and only included a mention of Rogers' theorem after being alerted to it by Tenenbaum. So it is perhaps not inconceivable that even Erdos did not recall Rogers' theorem when preparing his long paper of open questions with Graham in 1980.
(emphasis mine)
I think the value of LLM guided literature searches is pretty clear!
Both are precisely true. It is a better search engine than anything else -- which, while true, is something you won't realize unless you've used the non-free 'pro research' features from Google and/or OpenAI. And it can perform limited but increasingly-capable reasoning about what it finds before presenting the results to the user.
Note that no online Web search or tool usage at all was involved in the recent IMO results. I think a lot of people missed that little detail.