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.
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 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
It looks like these models work pretty well as natural language search engines and at connecting together dots of disparate things humans haven't done.
They're finding them very effective at literature search, and at autoformalization of human-written proofs.
Pretty soon, this is going to mean the entire historical math literature will be formalized (or, in some cases, found to be in error). Consider the implications of that for training theorem provers.
I think "pretty soon" is a serious overstatement. This does not take into account the difficulty in formalizing definitions and theorem statements. This cannot be done autonomously (or, it can, but there will be serious errors) since there is no way to formalize the "text to lean" process.
What's more, there's almost surely going to turn out to be a large amount of human generated mathematics that's "basically" correct, in the sense that there exists a formal proof that morally fits the arc of the human proof, but there's informal/vague reasoning used (e.g. diagram arguments, etc) that are hard to really formalize, but an expert can use consistently without making a mistake. This will take a long time to formalize, and I expect will require a large amount of human and AI effort.
This illustrates how unimportant this problem is. A prior solution did exist, but apparently nobody knew because people didn't really care about it. If progress can be had by simply searching for old solutions in the literature, then that's good evidence the supposed progress is imaginary. And this is not the first time this has happened with an Erdős problem.
A lot of pure mathematics seems to consist in solving neat logic puzzles without any intrinsic importance. Recreational puzzles for very intelligent people. Or LLMs.
There is still enormous value in cleaning up the long tail of somewhat important stuff. One of the great benefits of Claude Code to me is that smaller issues no longer rot in backlogs, but can be at least attempted immediately.
The difference is that Claude Code actually solves practical problems, but pure (as opposed to applied) mathematics doesn't. Moreover, a lot of pure mathematics seems to be not just useless, but also without intrinsic epistemic value, unlike science. See https://news.ycombinator.com/item?id=46510353
I’m an engineer, not a mathematician, so I definitely appreciate applied math more than I do abstract math. That said, that’s my personal preference and one of the reasons that I became an engineer and not a mathematician. Woking on nothing but theory would bore me to tears. But I appreciate that other people really love that and can approach pure math and see the beauty. And thank God that those people exist because they sometimes find amazing things that we engineers can use during the next turn of the technological crank. Instead of seeing pure math as useless, perhaps shift to seeing it as something wonderful for which we have not YET found a practical use.
Applications for pure mathematics can't necessarily be known until the underlying mathematics is solved.
Just because we can't imagine applications today doesn't mean there won't be applications in the future which depend on discoveries that are made today.
Well, read the linked comment. The possible future applications of useless science can't be known either. I still argue that it has intrinsic value apart from that, unlike pure mathematics.
You are not yet getting it I'm afraid. The point of the linked post was that, even assuming an equal degree of expected uselessness, scientific explanations have intrinsic epistemic value, while proving pure math theorems hasn't.
It's hard to know beforehand. Like with most foundational research.
My favorite example is number theory. Before cyptography came along it was pure math, an esoteric branch for just number nerds. defund Turns out, super applicable later on.
You’re confusing immediately useful with eventually useful. Pure maths has found very practical applications over the millennia - unless you don’t consider it pure anymore, at which point you’re just moving goalposts.
You are confusing that. The biggest advancements in science are the result of the application of leading-edge pure math concepts to physical problems. Netwonian physics, relativistic physics, quantum field theory, Boolean computing, Turing notions of devices for computability, elliptic-curve cryptography, and electromagnetic theory all derived from the practical application of what was originally abstract math play.
Among others.
Of course you never know which math concept will turn out to be physically useful, but clearly enough do that it's worth buying conceptual lottery tickets with the rest.
Just to throw in another one, string theory was practically nothing but a basic research/pure research program unearthing new mathematical objects which drove physics research and vice versa. And unfortunately for the haters, string theory has borne real fruit with holography, producing tools for important predictions in plasma physics and black hole physics among other things. I feel like culture hasn't caught up to the fact that holography is now the gold rush frontier that has everyone excited that it might be our next big conceptual revolution in physics.
There is a difference between inventing/axiomatizing new mathematical theories and proving conjectures. Take the Riemann hypothesis (the big daddy among the pure math conjectures), and assume we (or an LLM) prove it tomorrow. How high do you estimate the expected practical usefulness of that proof?
That's an odd choice, because prime numbers routinely show up in important applications in cryptography. To actually solve RH would likely involve developing new mathematical tools which would then be brought to bear on deployment of more sophisticated cryptography. And solving it would be valuable in its own right, a kind of mathematical equivalent to discovering a fundamental law in physics which permanently changes what is known to be true about the structure of numbers.
Ironically this example turns out to be a great object lesson in not underestimating the utility of research based on an eyeball test. But it shouldn't even have to have any intuitively plausible payoff whatsoever in order to justify it. The whole point is that even if a given research paradigm completely failed the eyeball test, our attitude should still be that it very well could have practical utility, and there are so many historical examples to this effect (the other commenter already gave several examples, and the right thing to do would have been acknowledge them), and besides I would argue they still have the same intrinsic value that any and all knowledge has.
It shows that a 'llm' can now work on issues like this today and tomorrow it can do even more.
Don't be so ignorant. A few years ago NO ONE could have come up with something so generic as an LLM which will help you to solve this kind of problems and also create text adventures and java code.
Its not just verbose—it's almost a novel. Parent either cooked and capped, or has managed to perfectly emulate the patterns this parrot is stochastically known best for. I liked the pro human vibe if anything.
That’s just the internet. Detecting sarcasm requires a lot of context external to the content of any text. In person some of that is mitigated by intonation, facial expressions, etc. Typically it also requires that the the reader is a native speaker of the language or at least extremely proficient.
regardless of if this text was written by an LLM or a human, it is still slop,with a human behind it just trying to wind people up . If there is a valid point to be made , it should be made, briefly.
If the point was triggering a reply, the length and sarcasm certainly worked.
I agree brevity is always preferred. Making a good point while keeping it brief is much harder than rambling on.
But length is just a measure, quality determines if I keep reading. If a comment is too long, I won’t finish reading it. If I kept reading, it wasn’t too long.
> This is a relief, honestly. A prior solution exists now, which means the model didn’t solve anything at all. It just regurgitated it from the internet, which we can retroactively assume contained the solution in spirit, if not in any searchable or known form. Mystery resolved.
Vs
> Interesting that in Terrance Tao's words: "though the new proof is still rather different from the literature proof)"
I suspect this is AI generated, but it’s quite high quality, and doesn’t have any of the telltale signs that most AI generated content does. How did you generate this? It’s great.
Their comments are full of "it's not x, it's y" over and over. Short pithy sentences. I'm quite confident it's AI written, maybe with a more detailed prompt than the average
And with enough motivated reasoning, you can find AI vibes in almost every comment you don’t agree with.
For better or worse, I think we might have to settle on “human-written until proven otherwise”, if we don’t want to throw “assume positive intent” out the window entirely on this site.
Dude is swearing up and down that they came up with the text on their own. I agree with you though, it reeks of LLMs. The only alternative explanation is that they use LLMs so much that they’ve copied the writing style.
I’m confused by this. I still see this kind of phrasing in LLM generated content, even as recent as last week (using Gemini, if that matters). Are you saying that LLMs do not generate text like this, or that it’s now possible to get text that doesn’t contain the telltale “its not X, it’s Y”?
I wouldn't know how to prove to you otherwise other then to tell you that I have seen these tools show incorrect results for both AI generated text and human written text.
It's bizarre. The same account was previously arguing in favor of emergent reasoning abilities in another thread ( https://news.ycombinator.com/item?id=46453084 ) -- I voted it up, in fact! Turing test failed, I guess.
We need a name for the much more trivial version of the Turing test that replaces "human" with "weird dude with rambling ideas he clearly thinks are very deep"
I'm pretty sure it's like "can it run DOOM" and someone could make an LLM that passes this that runs on an pregnancy test
I mean.. LLMs have hit a pretty hard wall a while ago, with the only solution being throwing monstrous compute at eking out the remaining few percent improvement (real world, not benchmarks). That's not to mention hallucinations / false paths being a foundational problem.
LLMs will continue to get slightly better in the next few years, but mainly a lot more efficient. Which will also mean better and better local models. And grounding might get better, but that just means less wrong answers, not better right answers.
So no need for doomerism. The people saying LLMs are a few years away from eating the world are either in on the con or unaware.
Can anyone give a little more color on the nature of Erdos problems? Are these problems that many mathematicians have spend years tackling with no result? Or do some of the problems evade scrutiny and go un-attempted for most of the time?
EDIT:
After reading a link someone else posted to Terrance Tao's wiki page, he has a paragraph that somewhat answers this question:
> Erdős problems vary widely in difficulty (by several orders of magnitude), with a core of very interesting, but extremely difficult problems at one end of the spectrum, and a "long tail" of under-explored problems at the other, many of which are "low hanging fruit" that are very suitable for being attacked by current AI tools. Unfortunately, it is hard to tell in advance which category a given problem falls into, short of an expert literature review. (However, if an Erdős problem is only stated once in the literature, and there is scant record of any followup work on the problem, this suggests that the problem may be of the second category.)
Erdos was an incredibly prolific mathematician, and one of his quirks is that he liked to collect open problems and state new open problems as a challenge to the field. Many of the problems he attached bounties to, from $5 to $10,000.
The problems are a pretty good metric for AI, because the easiest ones at least meet the bar of "a top mathematician didn't know how to solve this off the top of his head" and the hardest ones are major open problems. As AI progresses, we will see it slowly climb the difficulty ladder.
Don't feel bad for being out of the loop.
The author and Tao did not care enough about erdos problem to realize the proof was published by erdos himself.
So you never cared enough and neither did they.
But they care about about screaming LLMs breakthrough on fediverse and twitter.
This is bad faith. Erdos was an incredibly prolific mathematician, it is unreasonable to expect anyone to have memorized his entire output. Yet, Tao knows enough about Erdos to know which mathematical techniques he regularly used in his proofs.
From the forum thread about Erdos problem 281:
> I think neither the Birkhoff ergodic theorem nor the Hardy-Littlewood maximal inequality, some version of either was the key ingredient to unlock the problem, were in the regular toolkit of Erdos and Graham (I'm sure they were aware of these tools, but would not instinctively reach for them for this sort of problem). On the other hand, the aggregate machinery of covering congruences looks relevant (even though ultimately it turns out not to be), and was very much in the toolbox of these mathematicians, so they could have been misled into thinking this problem was more difficult than it actually was due to a mismatch of tools.
> I would assess this problem as safely within reach of a competent combinatorial ergodic theorist, though with some thought required to figure out exactly how to transfer the problem to an ergodic theory setting. But it seems the people who looked at this problem were primarily expert in probabilistic combinatorics and covering congruences, which turn out to not quite be the right qualifications to attack this problem.
He's the most prolific and famous modern mathematician. I'm pretty sure that even if he'd never touched AI, he would be invited to more conferences than he could ever attend.
"Very nice! ... actually the thing that impresses me more than the proof method is the avoidance of errors, such as making mistakes with interchanges of limits or quantifiers (which is the main pitfall to avoid here). Previous generations of LLMs would almost certainly have fumbled these delicate issues.
...
I am going ahead and placing this result on the wiki as a Section 1 result (perhaps the most unambiguous instance of such, to date)"
The pace of change in math is going to be something to watch closely. Many minor theorems will fall. Next major milestone: Can LLMs generate useful abstractions?
Seems like the someone dug something up from the literature on this problem (see top comment on the erdosproblems.com thread)
"On following the references, it seems that the result in fact follows (after applying Rogers' theorem) from a 1936 paper of Davenport and Erdos (!), which proves the second result you mention. ... In the meantime, I am moving this problem to Section 2 on the wiki (though the new proof is still rather different from the literature proof)."
Personally, I'd prefer if the AI models would start with a proof of their own statements. Time and again, SOTA frontier models told me: "Now you have 100% correct code ready for production in enterprise quality." Then I run it and it crashes. Or maybe the AI is just being tongue-in-cheek?
Point in case: I just wanted to give z.ai a try and buy some credits. I used Firefox with uBlock and the payment didn't go through. I tried again with Chrome and no adblock, but now there is an error: "Payment Failed: p.confirmCardPayment is not a function." The irony is, that this is certainly vibe-coded with z.ai which tries to sell me how good they are but then not being able to conclude the sale.
And we will get lots more of this in the future. LLMs are a fantastic new technology, but even more fantastically over-hyped.
You get AIs to prove their code is correct in precisely the same ways you get humans to prove their code is correct. You make them demonstrate it through tests or evidence (screenshots, logs of successful runs).
FWIW, I just gave Deepseek the same prompt and it solved it too (much faster than the 41m of ChatGPT). I then gave both proofs to Opus and it confirmed their equivalence.
The answer is yes. Assume, for the sake of contradiction, that there exists an \(\epsilon > 0\) such that for every \(k\), there exists a choice of congruence classes \(a_1^{(k)}, \dots, a_k^{(k)}\) for which the set of integers not covered by the first \(k\) congruences has density at least \(\epsilon\).
For each \(k\), let \(F_k\) be the set of all infinite sequences of residues \((a_i)_{i=1}^\infty\) such that the uncovered set from the first \(k\) congruences has density at least \(\epsilon\). Each \(F_k\) is nonempty (by assumption) and closed in the product topology (since it depends only on the first \(k\) coordinates). Moreover, \(F_{k+1} \subseteq F_k\) because adding a congruence can only reduce the uncovered set. By the compactness of the product of finite sets, \(\bigcap_{k \ge 1} F_k\) is nonempty.
Choose an infinite sequence \((a_i) \in \bigcap_{k \ge 1} F_k\). For this sequence, let \(U_k\) be the set of integers not covered by the first \(k\) congruences, and let \(d_k\) be the density of \(U_k\). Then \(d_k \ge \epsilon\) for all \(k\). Since \(U_{k+1} \subseteq U_k\), the sets \(U_k\) are decreasing and periodic, and their intersection \(U = \bigcap_{k \ge 1} U_k\) has density \(d = \lim_{k \to \infty} d_k \ge \epsilon\). However, by hypothesis, for any choice of residues, the uncovered set has density \(0\), a contradiction.
Therefore, for every \(\epsilon > 0\), there exists a \(k\) such that for every choice of congruence classes \(a_i\), the density of integers not covered by the first \(k\) congruences is less than \(\epsilon\).
> I then gave both proofs to Opus and it confirmed their equivalence.
You could have just rubber-stamped it yourself, for all the mathematical rigor it holds. The devil is in the details, and the smallest problem unravels the whole proof.
"Since \(U_{k+1} \subseteq U_k\), the sets \(U_k\) are decreasing and periodic, and their intersection \(U = \bigcap_{k \ge 1} U_k\) has density \(d = \lim_{k \to \infty} d_k \ge \epsilon\)."
Is this enough? Let $U_k$ be the set of integers such that their remainder mod 6^n is greater or equal to 2^n for all 1<n<k. Density of each $U_k$ is more than 1/2 I think but not the intersection (empty) right?
Indeed. Your sets are decreasing periodic of density always greater than the product from k=1 to infinity of (1-(1/3)^k), which is about 0.56, yet their intersection is null.
This would all be a fairly trivial exercise in diagonalization if such a lemma as implied by Deepseek existed.
(Edit: The bounding I suggested may not be precise at each level, but it is asymptotically the limit of the sequence of densities, so up to some epsilon it demonstrates the desired counterexample.)
I don't see what's related there but anyway unless you have access to information from within OpenAI I don't see how you can claim what was or wasn't in the training data of ChatGPT 5.2 Pro.
On the contrary for DeepSeek you could but not for a non open model.
I find it interesting that, as someone utterly unfamiliar with ergodic theory, Dini’s theorem, etc, I find Deepseek’s proof somewhat comprehensible, whereas I do not find GPT-5.2’s proof comprehensible at all. I suspect that I’d need to delve into the terminology in the GPT proof if I tried to verify Deepseek’s, so maybe GPT’s is being more straightforward about the underlying theory it relies on?
There was a post about Erdős 728 being solved with Harmonic’s Aristotle a little over a week ago [1] and that seemed like a good example of using state-of-the-art AI tech to help increase velocity in this space.
I’m not sure what this proves. I dumped a question into ChatGPT 5.2 and it produced a correct response after almost an hour [2]?
Okay? Is it repeatable? Why did it come up with this solution? How did it come up with the connections in its reasoning? I get that it looks correct and Tao’s approval definitely lends credibility that it is a valid solution, but what exactly is it that we’ve established here? That the corpus that ChatGPT 5.2 was trained on is better tuned for pure math?
I’m just confused what one is supposed to take away from this.
I'm actually not sure what the right attribution method would be. I'd lean towards single line on acknowledgements? Because you can use it for example @ every lemma during brainstorming but it's unclear the right convention is to thank it at every lemma...
Anecdotally, I, as a math postdoc, think that GPT 5.2 is much stronger qualitatively than anything else I've used. Its rate of hallucinations is low enough that I don't feel like the default assumption of any solution is that it is trying to hide a mistake somewhere. Compared with Gemini 3 whose failure mode when it can't solve something is always to pretend it has a solution by "lying"/ omitting steps/making up theorems etc... GPT 5.2 usually fails gracefully and when it makes a mistake it more often than not can admit it when pointed out.
I guess the first question I have is if these problems solved by LLMs are just low-hanging fruit that human researchers either didn't get around to or show much interest in - or if there's some actual beef here to the idea that LLMs can independently conduct original research and solve hard problems.
That's the first warning from the wiki : <<Erdős problems vary widely in difficulty (by several orders of magnitude), with a core of very interesting, but extremely difficult problems at one end of the spectrum, and a "long tail" of under-explored problems at the other, many of which are "low hanging fruit" that are very suitable for being attacked by current AI tools.>> https://github.com/teorth/erdosproblems/wiki/AI-contribution...
There is still value on letting these LLMs loose on the periphery and knocking out all the low hanging fruit humanity hasn’t had the time to get around to. Also, I don’t know this, but if it is a problem on Erdos I presume people have tried to solve it atleast a little bit before it makes it to the list.
Is there though? If they are "solved" (as in the tickbox mark them as such, through a validation process, e.g. another model confirming, formal proof passing, etc) but there is no human actually learning from them, what's the benefit? Completing a list?
I believe the ones that are NOT studied are precisely because they are seen as uninteresting. Even if they were to be solved in an interesting way, if nobody sees the proof because they are just too many and they are again not considered valuable then I don't see what is gained.
Out of curiosity why has the LLM math solving community been focused on the Erdos problems over other open problems? Are they of a certain nature where we would expect LLMs to be especially good at solving them?
I guess they are at a difficulty where it's not too hard (unlike millennium prize problems), is fairly tightly scoped (unlike open ended research), and has some gravitas (so it's not some obscure theorem that's only unproven because of it's lack of noteworthiness).
I actually don't think the reason is that they are easier than other open math problems. I think it's more that they are "elementary" in the sense that the problems usually don't require a huge amount of domain knowledge to state.
I agree it's easier than Collatz. I just mean I am not sure it's much easier than many currently open questions which are less famous but need more machinery.
I was hoping there'd be more discussion about the model itself. I find the last couple of generations of Pro models fascinating.
Personally, I've been applying them to hard OCR problems. Many varied languages concurrently, wildly varying page structure, and poor scan quality; my dataset has all of these things. The models take 30 minutes a page, but the accuracy is basically 100% (it'll still striggle with perfectly-placed bits of mold). The next best model (Google's flagship) rests closer to 80%.
I'll be VERY intrigued to see what the next 2, 5, 10 years does to the price of this level of model.
I wonder if they tried Gemini. I think Gemini could have done better, as seen from my experiences with GPT and Gemini models on some simple geometry problems.
They never brothered to check erdos solution already published 90 years ago.
I am still confused about why erdos, who proposed the problem and the solution would consider this an unsolved problems, but this group of researchers would claim "ohh my god look at this breakthrough"
With a calculator I supply the arithmetic. It just executes it with no reasoning so im the solver. I can do the same with an LLM and still be the solver as long as it just follows my direction. Or I can give it a problem and let it reason and generate the arithmetic itself, in which case the LLM is effectively the solver. Thats why saying "I've solved X using only GPT" is ambiguous.
But thanks for the downvote in addition to your useless comment.
The LLMs that take 10 attempts to un-zero-width a <div>, telling me that every single change totally fixed the problem, are cracking the hardest math problems again.
This is crazy. It's clear that these models don't have human intelligence, but it's undeniable at this point that they have _some_ form of intelligence.
My take is that a huge part of human intelligence is pattern matching. We just didn’t understand how much multidimensional geometry influenced our matches
Yes, it could be that intelligence is essentially a sophisticated form of recursive, brute force pattern matching.
I'm beginning to think the Bitter Lesson applies to organic intelligence as well, because basic pattern matching can be implemented relatively simply using very basic mathematical operations like multiply and accumulate, and so it can scale with massive parallelization of relatively simple building blocks.
Intelligence is almost certainly a fundamentally recursive process.
The ability to think about your own thinking over and over as deeply as needed is where all the magic happens. Counterfactual reasoning occurs every time you pop a mental stack frame. By augmenting our stack with external tools (paper, computers, etc.), we can extend this process as far as it needs to go.
LLMs start to look a lot more capable when you put them into recursive loops with feedback from the environment. A trillion tokens worth of "what if..." can be expended without touching a single token in the caller's context. This can happen at every level as many times as needed if we're using proper recursive machinery. The theoretical scaling around this is extremely favorable.
I don't think it's accurate to describe LLMs as pattern matching. Prediction is the mechanism they use to ingest and output information, and they end up with a (relatively) deep model of the world under the hood.
> I don't think it's accurate to describe LLMs as pattern matching
I’m talking about the inference step, which uses tensor geometry arithmetic to find patterns in text. We don’t understand what those patterns are but it’s clear it’s doing some heavy lifting since llm inference is expressing logic and reasoning under the guise of our reductive “next token prediction”
The "pattern matching" perspective is true if you zoom in close enough, just like "protein reactions in water" is true for brains. But if you zoom out you see both humans and LLMs interact with external environments which provide opportunity for novel exploration. The true source of originality is not inside but in the environment. Making it be all about the model inside is a mistake, what matters more than the model is the data loop and solution space being explored.
"Pattern matching" is not sufficiently specified here for us to say if LLMs do pattern matching or not. E.g. we can say that an LLM predicts the next token because that token (or rather, its embedding) is the best "match" to the previous tokens, which form a path ("pattern") in embedding space. In this sense LLMs are most definitely pattern matching. Under other formulations of the term, they may not be (e.g. when pattern matching refers to abstraction or abstracting to actual logical patterns, rather than strictly semantic patterns).
Chess and Go have very restrictive rules. It seems a lot more obvious to me why a computer can beat a human at it. They have a huge advantage just by being able to calculate very deep lines in a very short time. I actually find it impressive for how long humans were able to beat computers at go. Math proofs seem a lot more open ended to me.
I don't think they will ever have human intelligence. It will always be an alien intelligence.
But I think the trend line unmistakably points to a future where it can be MORE intelligent than a human in exactly the colloquial way we define "more intelligent"
The fact that one of the greatest mathematicians alive has a page and is seriously bench marking this shows how likely he believes this can happen.
There's some nuance. IQ tests measure pattern matching and, in an underlying way, other facets of intelligence - memory, for example. How well can an LLM 'remember' a thing? Sometimes Claude will perform compaction when its context window reaches 200k "tokens" then it seems a little colder to me, but maybe that's just my imagination. I'm kind of a "power user".
what are you referring to? LLMs are neural networks at their core and the most simple versions of neural networks are all about reproducing patterns observed during training
You need to understand the difference between general matching and pattern matching. Maybe should have read more older AI books. A LLM is a general fuzzy matcher. A pattern matcher is an exact matcher using an abstract language, the "pattern". A general matcher uses a distance function instead, no pattern needed.
Ie you want to find a subimage in a big image, possibly rotated, scaled, tilted, distorted, with noise. You cannot do that with a pattern matcher, but you can do that with a matcher, such as a fuzzy matcher, a LLM.
You want to find a go position on a go board. A LLM is perfect for that, because you don't need to come up with a special language to describe go positions (older chess programs did that), you just train the model if that position is good or bad, and this can be fully automated via existing literature and later by playing against itself. You train the matcher not via patterns but a function (win or loose).
As someone who doesn't understand this shit, and how it's always the experts who fiddle the LLMs to get good outputs, it feels natural to attribute the intelligence to the operator (or the training set), rather than the LLM itself.
how did they do it? Was a human using the chat interface? Did they just type out the problem and immediately on the first reply received a complete solution (one-shot) or what was the human's role? What was ChatGPT's thinking time?
very interesting. ChatGPT reasoned for 41 minutes about it! Also, this was one-shot - i.e. ChatGPT produced its complete proof with a single prompt and no more replies by the human, (rather than a chat where the human further guided it.)
Funny seeing silicon valley bros commenting "you're on fire!" to Neel when it appears he copied and pasted the problem verbatim into chatGPT and it did literally all the other work here
I have 15 years of software engineering experience across some top companies. I truly believe that ai will far surpass human beings at coding, and more broadly logic work. We are very close
HN will be the last place to admit it; people here seem to be holding out with the vague 'I tried it and it came up with crap'. While many of us are shipping software without touching (much) code anymore. I have written code for over 40 years and this is nothing like no-code or whatever 'replacing programmers' before, this is clearly different judging from the people who cannot code with a gun to their heads but still are shipping apps: it does not really matter if anyone believes me or not. I am making more money than ever with fewer people than ever delivering more than ever.
We are very close.
(by the way; I like writing code and I still do for fun)
Both can be correct : you might be making a lot of money using the latest tools while others who work on very different problems have tried the same tools and it's just not good enough for them.
The ability to make money proves you found a good market, it doesn't prove that the new tools are useful to others.
> holding out with the vague 'I tried it and it came up with crap'
Isn't that a perfectly reasonable metric? The topic has been dominated by hype for at least the past 5 if not 10 years. So when you encounter the latest in a long line of "the future is here the sky is falling" claims, where every past claim to date has been wrong, it's natural to try for yourself, observe a poor result, and report back "nope, just more BS as usual".
If the hyped future does ever arrive then anyone trying for themselves will get a workable result. It will be trivially easy to demonstrate that naysayers are full of shit. That does not currently appear to be the case.
Wasn't transformer 2017? There's been constant AI hype since at least that far back and it's only gotten worse.
If I release a claim once a month that armageddon will happen next month, and then after 20 years it finally does, are all of my past claims vindicated? Or was I spewing nonsense the entire time? What if my claim was the next big pandemic? The next 9.0 earthquake?
Transformers was 2017 and it had some implications on translation (which were in no way overstated), but it took GPT-2 and 3 to kick it off in earnest and the real hype machine started with ChatGPT.
What you are doing however is dismissing the outrageous progress on NLP and by extension code generation of the last few years just because people over hype it.
People over hyped the Internet in the early 2000s, yet here we are.
Well I've been seeing an objectionable amount of what I consider to be hype since at least transformers.
I never dismissed the actual verifiable progress that has occurred. I objected specifically to the hype. Are you sure you're arguing with what I actually said as opposed to some position that you've imagined that I hold?
> People over hyped the Internet in the early 2000s, yet here we are.
And? Did you not read the comment you are replying to? If I make wild predictions and they eventually pan out does that vindicate me? Or was I just spewing nonsense and things happened to work out?
"LLMs will replace developers any day now" is such a claim. If it happens a month from now then you can say you were correct. If it doesn't then it was just hype and everyone forgets about it. Rinse and repeat once every few months and you have the current situation.
I don't dispute that the situation is rapidly evolving. It is certainly possible that we could achieve AGI in the near future. It is also entirely possible that we might not. Claims such as that AGI is close or that we will soon be replacing developers entirely are pure hype.
When someone says something to the effect of "LLMs are on the verge of replacing developers any day now" it is perfectly reasonable to respond "I tried it and it came up with crap". If we were actually near that point you wouldn't have gotten crap back when you tried it for yourself.
There's a big difference between "I tried it and it produced crap" and "it will replace developers entirely any day now"
People who use this stuff everyday know that people who are still saying "I tried it and it produced crap" just don't know how to use it correctly. Those developers WILL get replaced - by ones who know how to use the tool.
> Those developers WILL get replaced - by ones who know how to use the tool.
Now _that_ I would believe. But note how different "those who fail to adapt to this new tool will be replaced" is from "the vast majority will be replaced by this tool itself".
If someone had said that six (give or take) months ago I would have dismissed it as hype. But there have been at least a few decently well documented AI assisted projects done by veteran developers that have made the front page recently. Importantly they've shown clear and undeniable results as opposed to handwaving and empty aspirations. They've also been up front about the shortcomings of the new tool.
> I have 15 years of software engineering experience across some top companies. I truly believe that ai will far surpass human beings at coding, and more broadly logic work. We are very close
Coding was never the hard part of software development.
Getting the architecture mostly right, so it's easy to maintain and modify in the future is IMO hard part, but I find that this is where AI shines. I have 20 years of SWE experience (professional) and (10 hobby) and most of my AI use is for architecture and scaffolding first, code second.
They can only code to specification which is where even teams of humans get lost. Without much smarter architecture for AI (LLMs as is are a joke) that needle isn’t going to move.
Real HN comment right here. "LLMs are a joke" - maybe don't drink the anti-hype kool aid, you'll blind yourself to the capability space that's out there, even if it's not AGI or whatever.
This is no longer true, a prior solution has just been found[1], so the LLM proof has been moved to the Section 2 of Terence Tao's wiki[2].
[1] - https://www.erdosproblems.com/forum/thread/281#post-3325
[2] - https://github.com/teorth/erdosproblems/wiki/AI-contribution...
And even odder that the proof was by Erdos himself and yet he listed it as an open problem!
Carbon copy would mean over fitting
But honestly source = "a knuckle sandwich" would be appropriate here.
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.
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
I know that at least some LLM products explicitly check output for similarity to training data to prevent direct reproduction.
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.
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)
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.
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
Pretty soon, this is going to mean the entire historical math literature will be formalized (or, in some cases, found to be in error). Consider the implications of that for training theorem provers.
What's more, there's almost surely going to turn out to be a large amount of human generated mathematics that's "basically" correct, in the sense that there exists a formal proof that morally fits the arc of the human proof, but there's informal/vague reasoning used (e.g. diagram arguments, etc) that are hard to really formalize, but an expert can use consistently without making a mistake. This will take a long time to formalize, and I expect will require a large amount of human and AI effort.
But as far as we know, the proof it wrote is original. Tao himself noted that it’s very different from the other proof (which was only found now).
That’s so far removed from a “search engine” that the term is essentially nonsense in this context.
A lot of pure mathematics seems to consist in solving neat logic puzzles without any intrinsic importance. Recreational puzzles for very intelligent people. Or LLMs.
Just because we can't imagine applications today doesn't mean there won't be applications in the future which depend on discoveries that are made today.
https://www.reddit.com/r/math/comments/dfw3by/is_there_any_e...
My favorite example is number theory. Before cyptography came along it was pure math, an esoteric branch for just number nerds. defund Turns out, super applicable later on.
Among others.
Of course you never know which math concept will turn out to be physically useful, but clearly enough do that it's worth buying conceptual lottery tickets with the rest.
Ironically this example turns out to be a great object lesson in not underestimating the utility of research based on an eyeball test. But it shouldn't even have to have any intuitively plausible payoff whatsoever in order to justify it. The whole point is that even if a given research paradigm completely failed the eyeball test, our attitude should still be that it very well could have practical utility, and there are so many historical examples to this effect (the other commenter already gave several examples, and the right thing to do would have been acknowledge them), and besides I would argue they still have the same intrinsic value that any and all knowledge has.
Don't be so ignorant. A few years ago NO ONE could have come up with something so generic as an LLM which will help you to solve this kind of problems and also create text adventures and java code.
Evidence shows otherwise: Despite the "20x" length, many people actually missed the point.
I agree brevity is always preferred. Making a good point while keeping it brief is much harder than rambling on.
But length is just a measure, quality determines if I keep reading. If a comment is too long, I won’t finish reading it. If I kept reading, it wasn’t too long.
Vs
> Interesting that in Terrance Tao's words: "though the new proof is still rather different from the literature proof)"
I guess this is the end of the human internet
"Glorified Google search with worse footnotes" what on earth does that mean?
AI has a distinct feel to it
For better or worse, I think we might have to settle on “human-written until proven otherwise”, if we don’t want to throw “assume positive intent” out the window entirely on this site.
It wasn't AI generated. But if it was, there is currently no way for anyone to tell the difference.
This is false. There are many human-legible signs, and there do exist fairly reliable AI detection services (like Pangram).
Negative feedback is the original "all you need."
You're lying: https://www.pangram.com/history/94678f26-4898-496f-9559-8c4c...
Not that I needed pangram to tell me that, it's obvious slop.
(edit: fixed link)
I'm pretty sure it's like "can it run DOOM" and someone could make an LLM that passes this that runs on an pregnancy test
LLMs will continue to get slightly better in the next few years, but mainly a lot more efficient. Which will also mean better and better local models. And grounding might get better, but that just means less wrong answers, not better right answers.
So no need for doomerism. The people saying LLMs are a few years away from eating the world are either in on the con or unaware.
The only possible explanation is people say things they don't believe out of FUD. Literally the only one.
EDIT: After reading a link someone else posted to Terrance Tao's wiki page, he has a paragraph that somewhat answers this question:
> Erdős problems vary widely in difficulty (by several orders of magnitude), with a core of very interesting, but extremely difficult problems at one end of the spectrum, and a "long tail" of under-explored problems at the other, many of which are "low hanging fruit" that are very suitable for being attacked by current AI tools. Unfortunately, it is hard to tell in advance which category a given problem falls into, short of an expert literature review. (However, if an Erdős problem is only stated once in the literature, and there is scant record of any followup work on the problem, this suggests that the problem may be of the second category.)
from here: https://github.com/teorth/erdosproblems/wiki/AI-contribution...
The problems are a pretty good metric for AI, because the easiest ones at least meet the bar of "a top mathematician didn't know how to solve this off the top of his head" and the hardest ones are major open problems. As AI progresses, we will see it slowly climb the difficulty ladder.
This is bad faith. Erdos was an incredibly prolific mathematician, it is unreasonable to expect anyone to have memorized his entire output. Yet, Tao knows enough about Erdos to know which mathematical techniques he regularly used in his proofs.
From the forum thread about Erdos problem 281:
> I think neither the Birkhoff ergodic theorem nor the Hardy-Littlewood maximal inequality, some version of either was the key ingredient to unlock the problem, were in the regular toolkit of Erdos and Graham (I'm sure they were aware of these tools, but would not instinctively reach for them for this sort of problem). On the other hand, the aggregate machinery of covering congruences looks relevant (even though ultimately it turns out not to be), and was very much in the toolbox of these mathematicians, so they could have been misled into thinking this problem was more difficult than it actually was due to a mismatch of tools.
> I would assess this problem as safely within reach of a competent combinatorial ergodic theorist, though with some thought required to figure out exactly how to transfer the problem to an ergodic theory setting. But it seems the people who looked at this problem were primarily expert in probabilistic combinatorics and covering congruences, which turn out to not quite be the right qualifications to attack this problem.
"Very nice! ... actually the thing that impresses me more than the proof method is the avoidance of errors, such as making mistakes with interchanges of limits or quantifiers (which is the main pitfall to avoid here). Previous generations of LLMs would almost certainly have fumbled these delicate issues.
...
I am going ahead and placing this result on the wiki as a Section 1 result (perhaps the most unambiguous instance of such, to date)"
The pace of change in math is going to be something to watch closely. Many minor theorems will fall. Next major milestone: Can LLMs generate useful abstractions?
"On following the references, it seems that the result in fact follows (after applying Rogers' theorem) from a 1936 paper of Davenport and Erdos (!), which proves the second result you mention. ... In the meantime, I am moving this problem to Section 2 on the wiki (though the new proof is still rather different from the literature proof)."
Point in case: I just wanted to give z.ai a try and buy some credits. I used Firefox with uBlock and the payment didn't go through. I tried again with Chrome and no adblock, but now there is an error: "Payment Failed: p.confirmCardPayment is not a function." The irony is, that this is certainly vibe-coded with z.ai which tries to sell me how good they are but then not being able to conclude the sale.
And we will get lots more of this in the future. LLMs are a fantastic new technology, but even more fantastically over-hyped.
The answer is yes. Assume, for the sake of contradiction, that there exists an \(\epsilon > 0\) such that for every \(k\), there exists a choice of congruence classes \(a_1^{(k)}, \dots, a_k^{(k)}\) for which the set of integers not covered by the first \(k\) congruences has density at least \(\epsilon\).
For each \(k\), let \(F_k\) be the set of all infinite sequences of residues \((a_i)_{i=1}^\infty\) such that the uncovered set from the first \(k\) congruences has density at least \(\epsilon\). Each \(F_k\) is nonempty (by assumption) and closed in the product topology (since it depends only on the first \(k\) coordinates). Moreover, \(F_{k+1} \subseteq F_k\) because adding a congruence can only reduce the uncovered set. By the compactness of the product of finite sets, \(\bigcap_{k \ge 1} F_k\) is nonempty.
Choose an infinite sequence \((a_i) \in \bigcap_{k \ge 1} F_k\). For this sequence, let \(U_k\) be the set of integers not covered by the first \(k\) congruences, and let \(d_k\) be the density of \(U_k\). Then \(d_k \ge \epsilon\) for all \(k\). Since \(U_{k+1} \subseteq U_k\), the sets \(U_k\) are decreasing and periodic, and their intersection \(U = \bigcap_{k \ge 1} U_k\) has density \(d = \lim_{k \to \infty} d_k \ge \epsilon\). However, by hypothesis, for any choice of residues, the uncovered set has density \(0\), a contradiction.
Therefore, for every \(\epsilon > 0\), there exists a \(k\) such that for every choice of congruence classes \(a_i\), the density of integers not covered by the first \(k\) congruences is less than \(\epsilon\).
\boxed{\text{Yes}}
You could have just rubber-stamped it yourself, for all the mathematical rigor it holds. The devil is in the details, and the smallest problem unravels the whole proof.
Is this enough? Let $U_k$ be the set of integers such that their remainder mod 6^n is greater or equal to 2^n for all 1<n<k. Density of each $U_k$ is more than 1/2 I think but not the intersection (empty) right?
This would all be a fairly trivial exercise in diagonalization if such a lemma as implied by Deepseek existed.
(Edit: The bounding I suggested may not be precise at each level, but it is asymptotically the limit of the sequence of densities, so up to some epsilon it demonstrates the desired counterexample.)
https://news.ycombinator.com/item?id=46664976
On the contrary for DeepSeek you could but not for a non open model.
It says that the OpenAI proof is a different one from the published one in the literature.
Whereas whether the Deepseek proof is the same as the published one, I dont know enough of the math to judge.
That was what I meant.
I've "solved" many math problems with LLMs, with LLMs giving full confidence in subtly or significantly incorrect solutions.
I'm very curious here. The Open AI memory orders and claims about capacity limits restricting access to better models are interesting too.
I’m not sure what this proves. I dumped a question into ChatGPT 5.2 and it produced a correct response after almost an hour [2]?
Okay? Is it repeatable? Why did it come up with this solution? How did it come up with the connections in its reasoning? I get that it looks correct and Tao’s approval definitely lends credibility that it is a valid solution, but what exactly is it that we’ve established here? That the corpus that ChatGPT 5.2 was trained on is better tuned for pure math?
I’m just confused what one is supposed to take away from this.
[1] https://news.ycombinator.com/item?id=46560445
[2] https://chatgpt.com/share/696ac45b-70d8-8003-9ca4-320151e081...
One wonders if some professional mathematicians are instead choosing to publish LLM proofs without attribution for career purposes.
"This LLM is kinda dumb in the thing I'm an expert in"
Anecdotally, I, as a math postdoc, think that GPT 5.2 is much stronger qualitatively than anything else I've used. Its rate of hallucinations is low enough that I don't feel like the default assumption of any solution is that it is trying to hide a mistake somewhere. Compared with Gemini 3 whose failure mode when it can't solve something is always to pretend it has a solution by "lying"/ omitting steps/making up theorems etc... GPT 5.2 usually fails gracefully and when it makes a mistake it more often than not can admit it when pointed out.
I believe the ones that are NOT studied are precisely because they are seen as uninteresting. Even if they were to be solved in an interesting way, if nobody sees the proof because they are just too many and they are again not considered valuable then I don't see what is gained.
Personally, I've been applying them to hard OCR problems. Many varied languages concurrently, wildly varying page structure, and poor scan quality; my dataset has all of these things. The models take 30 minutes a page, but the accuracy is basically 100% (it'll still striggle with perfectly-placed bits of mold). The next best model (Google's flagship) rests closer to 80%.
I'll be VERY intrigued to see what the next 2, 5, 10 years does to the price of this level of model.
I would love to know which concepts are active in the deeper layers of the model while generating the solution.
Is there a concept of “epsilon” or “delta”?
What are their projections on each other?
They never brothered to check erdos solution already published 90 years ago. I am still confused about why erdos, who proposed the problem and the solution would consider this an unsolved problems, but this group of researchers would claim "ohh my god look at this breakthrough"
But thanks for the downvote in addition to your useless comment.
> the best way to find a previous proof of a seemingly open problem on the internet is not to ask for it; it's to post a new proof
I'm beginning to think the Bitter Lesson applies to organic intelligence as well, because basic pattern matching can be implemented relatively simply using very basic mathematical operations like multiply and accumulate, and so it can scale with massive parallelization of relatively simple building blocks.
The ability to think about your own thinking over and over as deeply as needed is where all the magic happens. Counterfactual reasoning occurs every time you pop a mental stack frame. By augmenting our stack with external tools (paper, computers, etc.), we can extend this process as far as it needs to go.
LLMs start to look a lot more capable when you put them into recursive loops with feedback from the environment. A trillion tokens worth of "what if..." can be expended without touching a single token in the caller's context. This can happen at every level as many times as needed if we're using proper recursive machinery. The theoretical scaling around this is extremely favorable.
I’m talking about the inference step, which uses tensor geometry arithmetic to find patterns in text. We don’t understand what those patterns are but it’s clear it’s doing some heavy lifting since llm inference is expressing logic and reasoning under the guise of our reductive “next token prediction”
But I think the trend line unmistakably points to a future where it can be MORE intelligent than a human in exactly the colloquial way we define "more intelligent"
The fact that one of the greatest mathematicians alive has a page and is seriously bench marking this shows how likely he believes this can happen.
Ie you want to find a subimage in a big image, possibly rotated, scaled, tilted, distorted, with noise. You cannot do that with a pattern matcher, but you can do that with a matcher, such as a fuzzy matcher, a LLM.
You want to find a go position on a go board. A LLM is perfect for that, because you don't need to come up with a special language to describe go positions (older chess programs did that), you just train the model if that position is good or bad, and this can be fully automated via existing literature and later by playing against itself. You train the matcher not via patterns but a function (win or loose).
https://mehmetmars7.github.io/Erdosproblems-llm-hunter/probl...
https://chatgpt.com/share/696ac45b-70d8-8003-9ca4-320151e081...
We are very close.
(by the way; I like writing code and I still do for fun)
The ability to make money proves you found a good market, it doesn't prove that the new tools are useful to others.
Isn't that a perfectly reasonable metric? The topic has been dominated by hype for at least the past 5 if not 10 years. So when you encounter the latest in a long line of "the future is here the sky is falling" claims, where every past claim to date has been wrong, it's natural to try for yourself, observe a poor result, and report back "nope, just more BS as usual".
If the hyped future does ever arrive then anyone trying for themselves will get a workable result. It will be trivially easy to demonstrate that naysayers are full of shit. That does not currently appear to be the case.
If I release a claim once a month that armageddon will happen next month, and then after 20 years it finally does, are all of my past claims vindicated? Or was I spewing nonsense the entire time? What if my claim was the next big pandemic? The next 9.0 earthquake?
What you are doing however is dismissing the outrageous progress on NLP and by extension code generation of the last few years just because people over hype it.
People over hyped the Internet in the early 2000s, yet here we are.
I never dismissed the actual verifiable progress that has occurred. I objected specifically to the hype. Are you sure you're arguing with what I actually said as opposed to some position that you've imagined that I hold?
> People over hyped the Internet in the early 2000s, yet here we are.
And? Did you not read the comment you are replying to? If I make wild predictions and they eventually pan out does that vindicate me? Or was I just spewing nonsense and things happened to work out?
"LLMs will replace developers any day now" is such a claim. If it happens a month from now then you can say you were correct. If it doesn't then it was just hype and everyone forgets about it. Rinse and repeat once every few months and you have the current situation.
When someone says something to the effect of "LLMs are on the verge of replacing developers any day now" it is perfectly reasonable to respond "I tried it and it came up with crap". If we were actually near that point you wouldn't have gotten crap back when you tried it for yourself.
People who use this stuff everyday know that people who are still saying "I tried it and it produced crap" just don't know how to use it correctly. Those developers WILL get replaced - by ones who know how to use the tool.
Now _that_ I would believe. But note how different "those who fail to adapt to this new tool will be replaced" is from "the vast majority will be replaced by this tool itself".
If someone had said that six (give or take) months ago I would have dismissed it as hype. But there have been at least a few decently well documented AI assisted projects done by veteran developers that have made the front page recently. Importantly they've shown clear and undeniable results as opposed to handwaving and empty aspirations. They've also been up front about the shortcomings of the new tool.
Coding was never the hard part of software development.