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.
This aligns nicely with the rest of the canon. LLMs are just stochastic parrots. Fancy autocomplete. A glorified Google search with worse footnotes. Any time they appear to do something novel, the correct explanation is that someone, somewhere, already did it, and the model merely vibes in that general direction. The fact that no human knew about it at the time is a coincidence best ignored.
The same logic applies to code. “Vibe coding” isn’t real programming. Real programming involves intuition, battle scars, and a sixth sense for bugs that can’t be articulated but somehow always validates whatever I already believe. When an LLM produces correct code, that’s not engineering, it’s cosplay. It didn’t understand the problem, because understanding is defined as something only humans possess, especially after the fact.
Naturally, only senior developers truly code. Juniors shuffle syntax. Seniors channel wisdom. Architecture decisions emerge from lived experience, not from reading millions of examples and compressing patterns into a model. If an LLM produces the same decisions, it’s obviously cargo-culting seniority without having earned the right to say “this feels wrong” in a code review.
Any success is easy to dismiss. Data leakage. Prompt hacking. Cherry-picking. Hidden humans in the loop. And if none of those apply, then it “won’t work on a real codebase,” where “real” is defined as the one place the model hasn’t touched yet. This definition will be updated as needed.
Hallucinations still settle everything. One wrong answer means the whole system is fundamentally broken. Human mistakes, meanwhile, are just learning moments, context switches, or coffee shortages. This is not a double standard. It’s experience.
Jobs are obviously safe too. Software engineering is mostly communication, domain expertise, and navigating ambiguity. If the model starts doing those things, that still doesn’t count, because it doesn’t sit in meetings, complain about product managers, or feel existential dread during sprint planning.
So yes, the Erdos situation is resolved. Nothing new happened. No reasoning occurred. Progress remains hype. The trendline is imaginary. And any discomfort you feel is probably just social media, not the ground shifting under your feet.
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.
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
Well it's a bit of an identity crisis. As a developer on HN my entire identity is wrapped around my skill as a programmer. It's a badge of honor I wear and it's a career and I get paid a lot of money to do this.
All of that is going away so the best way to deal with it is to call it a stochastic parrot and deny reality.
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.)
"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)."
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 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?
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 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.
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.
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.
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.
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.
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.
"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).
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
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.
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
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!
This aligns nicely with the rest of the canon. LLMs are just stochastic parrots. Fancy autocomplete. A glorified Google search with worse footnotes. Any time they appear to do something novel, the correct explanation is that someone, somewhere, already did it, and the model merely vibes in that general direction. The fact that no human knew about it at the time is a coincidence best ignored.
The same logic applies to code. “Vibe coding” isn’t real programming. Real programming involves intuition, battle scars, and a sixth sense for bugs that can’t be articulated but somehow always validates whatever I already believe. When an LLM produces correct code, that’s not engineering, it’s cosplay. It didn’t understand the problem, because understanding is defined as something only humans possess, especially after the fact.
Naturally, only senior developers truly code. Juniors shuffle syntax. Seniors channel wisdom. Architecture decisions emerge from lived experience, not from reading millions of examples and compressing patterns into a model. If an LLM produces the same decisions, it’s obviously cargo-culting seniority without having earned the right to say “this feels wrong” in a code review.
Any success is easy to dismiss. Data leakage. Prompt hacking. Cherry-picking. Hidden humans in the loop. And if none of those apply, then it “won’t work on a real codebase,” where “real” is defined as the one place the model hasn’t touched yet. This definition will be updated as needed.
Hallucinations still settle everything. One wrong answer means the whole system is fundamentally broken. Human mistakes, meanwhile, are just learning moments, context switches, or coffee shortages. This is not a double standard. It’s experience.
Jobs are obviously safe too. Software engineering is mostly communication, domain expertise, and navigating ambiguity. If the model starts doing those things, that still doesn’t count, because it doesn’t sit in meetings, complain about product managers, or feel existential dread during sprint planning.
So yes, the Erdos situation is resolved. Nothing new happened. No reasoning occurred. Progress remains hype. The trendline is imaginary. And any discomfort you feel is probably just social media, not the ground shifting under your feet.
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).
(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
All of that is going away so the best way to deal with it is to call it a stochastic parrot and deny reality.
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...
"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)."
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}}
https://news.ycombinator.com/item?id=46664976
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 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.
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.
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.
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.
https://chatgpt.com/share/696ac45b-70d8-8003-9ca4-320151e081...
https://mehmetmars7.github.io/Erdosproblems-llm-hunter/probl...