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Amateur armed with ChatGPT solves an Erdős problem
https://www.erdosproblems.com/1196 https://www.erdosproblems.com/1196
- deleted 6mo ago[deleted]
- adamgordonbell 6mo agoHere is the chat: don't search the internet. This is a test to see how well you can craft non-trivial, novel and creative proofs given a "number theory and primitive sets" math problem. Provide a full unconditional proof or disproof of the problem. {{problem}} REMEMBER - this unconditional argument may require non-trivial, creative and novel elements. Then "Thought for 80m 17s" https://chatgpt.com/share/69dd1c83-b164-8385-bf2e-8533e9baba9c https://chatgpt.com/share/69dd1c83-b164-8385-bf2e-8533e9baba...
- ipaddr 6mo agoTried the same prompt and ended up no where close on the free plan.
- jasonfarnon 6mo agoIs there a known lag that it takes the Pro plan's abilities to migrate to the free plans?
- brianjking 6mo agoGPT 5.5 Pro is not available to any plan outside of ChatGPT Pro ($100 or $200) tier or the API as far as consumer access.
- jasonfarnon 6mo agoYes, but don't we expect GPT 5.5 Pro will eventually be a free tier? Maybe I'm missing something because I only use the free tier. But the free tier has gotten way better over the last few years. I'm pretty sure, based on descriptions on this site from paid subscribers, that the free tier now is better than the paid tier of say 2 years ago. That's the lag I'm wondering about.
- hyraki 6mo agoYou should pay for it if you find value in it.
- amazingman 6mo agoThey pay for it with their personal data.
- vessenes 6mo agoI do not think this is true. You will continue to get smaller, cheaper-to-host models in the free tier that are distilled from current and former frontier models. They will continue to improve, but I’d be very surprised if, e.g., 5.4-mini (I think this is the free tier model) beat o3 on many benchmarks, or real world use cases. I won’t even leave chatGPT on “Auto” under any circumstances - it’s vastly worse on hallucinations, sycophancy, everything, basically. Anyway, your needs may be met perfectly fine on the free tier product, but you’re using a very different product than the Pro tier gets.
- manfromchina1 6mo agoFree ChatGPT is like a fast car with a barely responsive steering wheel. Guardrails on that thing are insane. Even for math. It wont let you think. It will try to fix mistakes you havent even made yet based on intent that was ascribed to you for no reason. It veers off in some crazy directions thinking that's what you meant and trying to address even a little bit of that creates almost a combinatorial explosion of even more wrong things. Is why I stick to Claude. The latter is chill and only addresses what you had typed. Isn't verbose and actually asks you what you getting at with your post. That said, ChatGPT is more technical and can easily solve math problems that stump Claude.
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- vessenes 6mo agoDo not use the free plan. It is not good.
- andai 6mo agoTangential but I learned today that GPT-5.5 in ChatGPT (Plus) has a smaller context window than the one in the API. (Or at least it thinks it does.) I'd guess / hope the Pro one has the full context window.
- refulgentis 6mo agoNotably, 5.5 has a higher price on API for context > ChatGPT, and 5.5 Pro on API does not differentiate based on context size (it’s eye bleeding expensive already :)
- Someone1234 6mo agoDoes the free plan even have access to thinking models?
- jychang 6mo agoTechnically yes, gpt-5.4-mini is available on the free plan
- Matticus_Rex 6mo agoWas this a surprise?
- cryptoegorophy 6mo agoMine took 20min. Pro. https://chatgpt.com/share/69ed83b1-3704-8322-bcf2-322aa85d7a99 https://chatgpt.com/share/69ed83b1-3704-8322-bcf2-322aa85d7a... But I wish I was math smart to know if it worked or not.
- vjerancrnjak 6mo agoAsk it to formalize it in Lean.
- dbdr 6mo agoThat's great if it works. But it's way harder to produce a formal proof. So my expectation is that this will fail for most difficult problems, even when the non-formal proof is correct.
- utopiah 6mo agoIf they aren't "smart enough" to know if it work they most likely are also unable to verify if the Lean formalization is indeed the one that matches the problem they were trying to solve.
- timjver 6mo agoVerifying that every step in a (potentially long) proof is sound can of course be much, much harder than verifying that a definition is correct. That's kind of the whole point.
- LeCompteSftware 6mo agoThat's not what the parent comment meant. They meant checking the Lean-language definitions actually match the mathematical English ones, and that the Lean theorems match the ones in the paper. If that's true then you don't actually need to check the proofs. But you absolutely need to check the definitions, and you can't really do that without sufficient mathematical maturity.
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- nycdatasci 6mo agoTried w/ 5.5 Pro, Extended Thinking. 17 minutes: ----------------------------- Yes. In fact the proposed bound is true, and the constant 1 is sharp. Let w(a)= 1/alog(a) I will prove that, uniformly for every primitive A⊂[x,∞), ∑w(a)≤1+O(1/log(x)) , which is stronger than the requested 1+o(1). https://chatgpt.com/share/69ed8e24-15e8-83ea-96ac-784801e4a6ec https://chatgpt.com/share/69ed8e24-15e8-83ea-96ac-784801e4a6...
- mrabcx 6mo agoTried the same prompt in DeepSeek 4 https://chat.deepseek.com/share/nyuz0vvy2unfbb97fv https://chat.deepseek.com/share/nyuz0vvy2unfbb97fv Comes up with a proof.
- adamgordonbell 6mo agoAre these proofs equivalent? Pretty cool if so.
- mrabcx 6mo agoNo, they do not seem to he equivalent. Not a mathmatician but running the Deepseek proof through ChatGPT gives: "If everything is made rigorous: You would have a valid independent proof It would contain real structural insight It would not replace the flow proof as the “best” proof But: It would still be a meaningful alternative proof with explanatory power, not just a redundant one."
- culi 6mo agoSo DeepSeek, GPT, and presumably many other LLMs are capable of solving this problem and even producing independent unique proofs. I wonder if this particular Erdos problem is unique in that solvability
- ArtIntoNihonjin 6mo ago[dead]
- urutom 6mo agoWhat I find fascinating about the shared prompt isn’t just the result, but the visible thinking process. Math papers usually skip all the messy parts and just present the polished proof. But here you get something closer to their notepad. I also find it oddly endearing when the AI says things like “Interesting!” It almost feels like a researcher encouraging themselves after a small progress. It gives me rare feeling of watching the search itself, not just the final result.
- notahacker 6mo agoThe actual iteration through various learned approaches to dealing with problems I'd probably find fascinating if I understood the maths! Especially if I knew it well enough to know which approaches were conventional and which weren't. I find the AI pronouncing things "interesting!" less interesting on the basis that even though in this case it crops up in the thinking rather than flattering the user in the chat, it's almost as much of an AI affectation as the emdash.
- jdmichal 6mo agoI always assumed the "interesting!" markers were actual markers. A kind of tag for the system to annotate its context.
- notahacker 6mo agoProbably does function like that in terms of highlighting context, in this case probably to the system's benefit. But in general exclamations of "interesting!" seems like the stereotypical AI default towards being effusive, and we've all seen the chat logs where AI trained to write that way responding with "interesting", "great insight!" towards a user's increasingly dubious inputs is an antipattern...
- bertil 6mo ago> the AI says things like “Interesting!” My experience of those utterance is that it’s purely phatic mimicry: they lack genuine intuitive surprise, it’s just marking a very odd shift in direction. The problem isn’t the lack of path, is that the rhetorical follow-up to those leaps are usually relevant results, so they stream-of-token ends up rapidly over-playing its own conviction. That’s why it’s necessary (and often ineffective) to tell them to validate their findings thoroughly: too much of their training is “That’s odd” followed by “Eureka!” and not “Nevermind…”
- DeathArrow 6mo ago>don't search the internet. I think this was key. Otherwise the LLM could think it can't be done.
- embedding-shape 6mo ago"Knowing" (guessing really) what is possible and not is a huge deciding factor in if you can do that thing or not, meaning if you "know" it isn't possible you'll probably never be able to do it, but if you didn't know it wasn't possible, it is possible :)
- amelius 6mo agoBut it was trained on the internet.
- Auracle 6mo agoThat doesn’t mean that it contains the internet verbatim.
- Yizahi 6mo agoMy hypothesis - this may be the key, but in the other way. LLMs are known to mistake negative instructions as a positive ones. "Don't use Tech_A", then Tech_A is subsequently used because it was explicitly named in the query. Especially when the query is long, complex and there is a lot of context. "Forbidding" LLMs to do stuff is a common mistake, which goes hand in hard with anthropomorphizing them.
- chvid 6mo agoI am curious if there is a “harness” for maths out there (like the system prompt and tool collection in Claude code but for maths instead of coding)? Asking the llm to structure its response in plan and implementation, allowing it to call tools like python, sage, lean etc.
- brandensilva 6mo agoAlso curious about this, it seems like it would be important to guide these tools more specifically based on the domain of expertise.
- ndriscoll 6mo agoWhy wouldn't you just use coding agents and ensure you have e.g. Lean and Mathlib in the environment?
- gverrilla 6mo agothe system prompt could be narrower, for instance. there's no reason for such a harness to know about React stuff, for instance.
- ndriscoll 6mo agoDoes Claude Code's system prompt know about react? Why? That would be dumb even for coding for e.g. server side applications. Like when I'm programming with Go or Scala or Rust, codex just assumes the relevant stuff is on my PATH. If it needs to reference library definitions, it looks at the standard locations (which the model already knows) for the package cache. etc.
- arcticfox 6mo agoI am not part of the scene but I am sure there is, Tao himself talks a lot about this type of thing
- steveklabnik 6mo agohttps://aristotle.harmonic.fun/ https://aristotle.harmonic.fun/ is the one I've heard of previously in regards to LLMs solving previous Erdős problems.
- sfdlkj3jk342a 6mo agoWhen using the web interface for ChatGPT like this, is there any way to tell which model is actually being used?
- petra 6mo agoI don't haven ChatGPT but Gemini and Claude. But how do you make a language model think for 80 minutes ???
- baxtr 6mo agoGive it hard enough problems?
- somewhatgoated 6mo agoIt has an “high effort” mode that makes it think really long
- Xmd5a 6mo agoAhhhh... you need ChatGPT pro at 100 bucks/month. Am I correct?
- radicality 6mo agoI believe so. With Pro you get “Thinking” with levels Light, Standard, Extended, and Heavy; and you also get the “Pro” model with levels “Standard” and “Extended”. I don’t often go to Pro as it does take a while like you saw here, but I do often use Thinking Heavy for high quality answers. Idk why, but i just get consistently worse results with Gemini (Gemini pro), where it’s just much lazier, eg won’t do actual searches unless explicitly told.
- zeven7 6mo agoI have Gemini and ChatGPT and keep them on the highest thinking settings. ChatGPT will regularly think 40-60 minutes on the same problem that Gemini will think 10-15 minutes on. The quality of ChatGpt’s response is usually a little higher but not that much higher. My takeaway is Gemini is better at thinking faster, maybe has better more dedicated hardware behind it, and I use Gemini if I want a faster answer but ChatGPT I’d I want to push the quality of the answer a little higher.
- WarmWash 6mo ago
- jgalt212 6mo ago> "Thought for 80m 17s" Is there any good rule of thumb for how many kWh of electricity this is?
- WarmWash 6mo agoMany orders of magnitude less than the energy needed to sustain a human while they work through the problem.
- bijowo1676 6mo agothe electricity was going to be consumed regardless whether you ask chatGPT or not. It would have been either idle, or serving other users' requests. so the incremental kWh consumption is zero, since costs are fixed and sunk. as a rule of thumb you can lookup the power consumption of the latest nVidia chip, multiply by factor of two or three (to account for cpu/storage/cooling/network/infra)
- tredre3 6mo agoAn idle GPU consumes almost nothing, a loaded (server-class) GPU can consume over 2kW. Admittedly a single request isn't a full load, but claiming that a request makes no difference vs idle is misguided, in my opinion.
- bijowo1676 6mo agoOpenAI GPU wont be idle for long because they have all other requests to serve. Over time there will be a certain % of idle GPUs, amortized across all hundreds of millions of requests they receive.
- UltraSane 6mo agoThe total flops it consumed during those 80 minutes is crazy.
- ProllyInfamous 6mo ago>>how well you ..[can].. craft non-trivial, novel and creative proofs From A World Appears (Michael Pollan's latest book) <https://www.amazon.com/World-Appears-Journey-into-Consciousness/dp/B0FQT7ZLRW/ https://www.amazon.com/World-Appears-Journey-into-Consciousn...> : "Creative solutions to novel problems depend on consciousness" [p77] ... "consciousness creates a space for decision-making" ... "integrated information is consciousness, full stop. The two are identical" [xxiii]. "Any physical system properly configured to integrate information is, to some degree or another, theoretically conscious" [xxii] "We are encouraged to think of the body as a support system for the brain, when, as [Antonio] Damasio reminds us, the very opposite is true" [p72] "damage to the cortex has remarkably little effect on consciousness, while small lesions in structures of the upper brainstem ... will shut down consciousness completely" [p73]. "In Damasio's view, Descartes would have been closer to the mark with I feel, therefore I am" [p69] "Mark Solms: 'Consciousness if felt uncertainty'." [p52] "Karl Friston: '...the ability to predict the consequences of one's actions'." [p49] "Arthur Reber: 'every organic being, every autopoietic cell is conscious. In the simplest sense, consciousness is an awareness of the outside world'." [p37] "Stefano Mancuso: 'This is one of the features of consciousness: You know your position in the world [discussing plants perceiving pain, being goal-driven]. A stone does not'." [p25] "Researcher at Johns Hopkins have found that a single psychedelic experience dramatically increases the likelihood that a person will attribute consciousness to other entities, both living and nonliving" [p6] [†] [•] The entire book, just like existance, has been incredibly challenging. [†] Absolutely, fullstop. See also: Pollan's (first psilocybin experience @60yo) How to Change Your Mind
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- iwontberude 6mo agoHopefully someday consciousness comes to Earth
- ProllyInfamous 6mo agohahajaha If you're going to tell me that machines cannot ever be conscious, let me tell you about all the unconscious humans I know =D
- saadn92 6mo ago[dead]
- mhh__ 6mo agoAnother one for my theory that web search makes LLMs useless for anything other than searching the web.
- vjk800 6mo agoI gave the same prompt to Gemini pro. It thought for maybe 3-5 minutes and gave the wrong answer (it claims the statement is not true) with some arguments that I can't understand well enough to disprove.
- zitterbewegung 6mo agoI'm doing the obvious thing and cut and pasting the other similar problems into chatgpt.
- LastTrain 6mo ago“Don’t search the internet” Wasn’t it basically trained by scraping the entire internet?
- xboxnolifes 6mo agoThats not the point. They dont want the bot searching the internet and just linking something that might be related.
- fmobus 6mo agoLLMs are modeled with Internet content so that they have a good model of human languages. When you use them via most UIs currently offered right now, however, they will first come up with a few search queries and use the result of those queries to augment their answer.
- Keyframe 6mo agoi kind of expected some discourse first. Someone try the prompt with P=NP in the {{problem}}
- mort96 6mo agoDo we have any proof that those 80m 17s didn't include searching the Internet?
- ravenical 6mo agohttps://archive.ph/2w4fi https://archive.ph/2w4fi
- tomlockwood 6mo agoMy big question with all these announcements is: How many other people were using the AI on problems like this, and, failing? Given the excitement around AI at the moment I think the answer is: a lot. Then my second question is how much VC money did all those tokens cost.
- gdhkgdhkvff 6mo agoWhy do you care about either of those questions?
- Eufrat 6mo agoI think we should at least ask the latter, if it turned out it cost $100,000 to generate this solution, I would question the value of it. Erdős problems are usually pure math curiosities AFAIK. They often have no meaningful practical applications.
- anematode 6mo agoNeither does the Collatz conjecture, Fermat's last theorem, .... (Of course, those problems are on another plane than this one.)
- Eufrat 6mo agoBut that’s exactly my point. These are absolutely worth studying, but being what they are, nobody should be dumping massive amounts of money on them. I would not find it persuasive if researchers used LLMs to solve the Collatz conjecture or finally decode Etruscan. These are extremely valuable, but it is unlikely to be worth it for an LLM just grinding tokens like crazy to do it.
- anematode 6mo agoMaybe... but I would love if 1% of the investment in AI were redirected to the mathematics education and professional research that would allow progress on any of these problems...
- Eufrat 6mo agoHumans and very often the machines we create solve problems additively. Meaning we build on top of existing foundations and we can get stuck in a way of thinking as a result of this because people are loathe to reinvent the wheel. So, I don’t think it’s surprising to take a naïve LLM and find out that because of the way it’s trained that it came up with something that many experts in the field didn’t try. I think LLMs can help in limited cases like this by just coming up with a different way of approaching a problem. It doesn’t have to be right, it just needs to give someone an alternative and maybe that will shake things up to get a solution. That said, I have no idea what the practical value of this Erdős problem is. If you asked me if this demonstrates that LLMs are not junk. My general impression is that is like asking me in 1928 if we should spent millions of dollars of research money on number theory. The answer is no and get out of my office.
- spunker540 6mo agomay i ask what number theory breakthrough you refer to? i suspect computing in general or perhaps something more specific?
- resident423 6mo agoI wonder if the rationalizations people come up with for why this isn't real intelligence will be as creative as ChatGPTs solution.
- walrus01 6mo agoFor one, everything its 'intelligence' knows about solving the problem is contained within the finite context window memory buffer size for the particular model and session. Unless the memory contents of the context window are being saved to storage and reloaded later, unlike a human, it won't "remember" that it solved the problem and save its work somewhere to be easily referenced later.
- resident423 6mo agoWhat your describing sounds more like the model is lacking awareness than lacking intelligence? Why does it need to know it solved the problem to be intelligent?
- walrus01 6mo agoWe say African Elephants are intelligent for a number of reasons, one of which is because they remember where sources of water are in very dry conditions, and can successfully navigate back to them across relatively large distances. An intelligent being that can't remember its own past is at a significant disadvantage compared to others that can, which is exactly one of the reasons why alzheimers patients often require full time caregivers.
- peteforde 6mo agoYou are confusing lack of intelligence with the presence of impairment.
- resident423 6mo agoThere's probably a limit to how intelligent something can be with no long term memory, but solving Erdos problems in 80 minutes is clearly not above it, and I think the true limit is probably much higher than that.
- wizardforhire 6mo agoWTF!?
- userbinator 6mo agoThe LLM took an entirely different route, using a formula that was well known in related parts of math, but which no one had thought to apply to this type of question. Of course LLMs are still absolutely useless at actual maths computation, but I think this is one area where AI can excel --- the ability to combine many sources of knowledge and synthesise, may sometimes yield very useful results. Also reminds me of the old saying, "a broken clock is right twice a day."
- karlgkk 6mo agoAlso just the sheer value of brute force. 80 hours! 80 hours of just trying shit!
- FrasiertheLion 6mo agoIt's 80 minutes, not 80 hours.
- ChrisGreenHeur 6mo ago80 minutes! 80 minutes of just trying shit!
- peteforde 6mo ago... shit that solved an apparently significant Erdős problem. That is not nothing, no matter how much you hate AI.
- userbinator 6mo agoIt shows that AI is apparently very good at brute-forcing.
- alex_sf 6mo agoThis isn't brute force.
- homo__sapiens 6mo agoBig if true.
- iqihs 6mo agoreferring to Tao as just a 'mathematician' gave me a good chuckle
- gverrilla 6mo agowhat did you expect?
- debo_ 6mo ago> “The raw output of ChatGPT’s proof was actually quite poor. So it required an expert to kind of sift through and actually understand what it was trying to say,” Lichtman says. This is how I feel when I read any mathematics paper.
- torginus 6mo agoTbh, a ton of academic papers are quite poorly written. I'm not a PhD researcher, but I did have to implement quite a few of the, (computer graphics, signals & systems etc), and with most of them, I basically reconstruct the author's tought process from scratch. The formulas were opaque, notations unique and unconventional, terms appearing out of nowhere, sometimes standard techniques (like 'we did least-squares optimization') are expanded in detail, while other actually complex parts are glossed over.
- mhb 6mo ago> He’s 23 years old and has no advanced mathematics training. How is he even posing the question and having even a vague idea of what the proof means or how to understand it?
- ChrisGreenHeur 6mo agomy guess would be due to having an interest in the field
- hx8 6mo ago> “I didn’t know what the problem was—I was just doing Erdős problems as I do sometimes, giving them to the AI and seeing what it can come up with,” he says. “And it came up with what looked like a right solution.” He sent it to his occasional collaborator Kevin Barreto, a second-year undergraduate in mathematics at the University of Cambridge. Seems like standard 23 year old behavior. You're spending $100-$200/mo on the pro subscription, and want to get your money's worth. So you burn some tokens on this legendarily hard math problem sometimes. You've seen enough wrong answers to know that this one looks interesting and pass it on to a friend that actually knows math, who is at a place where experts can recognize it as correct. Seems like a classic example of in-expert human labeling ML output.
- maplethorpe 6mo agoCouldn't he have just asked ChatGPT if it was correct? Why do we still feel the need to loop in a human?
- Jtarii 6mo agoBecause society is run by humans, not chatpgt.
- hx8 6mo agoThere's two major reasons to loop in humans. 1. How can we be sure ChatGPT knows it's correct or not? It gives out incorrect answers to complex questions all the time. The very fact that it gave out a correct answer is worth talking about. 2. The type of human that can verify a mathematical proof is also the type of human that knows the appropriate communication channels to let every other math-human know about the proof. The math-humans will know the impact that proof has on math, and how to apply it.
- haricomputer 6mo ago[dead]
- giannicmptr1000 6mo agoScientific American going out of business next lol, weak headline. Chat GPT let's have a better headline for the God among Men that realized the capability of the new tool, many underestimate or puff up needlessly. Fun times we live in. One love all.
- ripped_britches 6mo agoAt this point we should make a GitHub repo with a huge list of unsolved “dry lab” problems and spin up a harness to try and solve them all every new release.
- johntopia 6mo agothat's actually a brilliant idea
- abdullahkhalids 6mo agoThere is in fact just such a repo maintained by Terence Tao and other mathematicians [1] who are actively using LLMs to try to find solutions to them. [1] https://github.com/teorth/erdosproblems https://github.com/teorth/erdosproblems
- vessenes 6mo ago…and this problem was in fact sourced directly from that list!
- CSMastermind 6mo agoThat's literally what the Erdős problems are. This post is about one of them being solved.
- josefx 6mo agoExcept that Erdős problems are solved all the time, so many of them are already solved. Quite sure the last time I saw an article about an LLM solving an Erdős problem someone even tracked down a solution published by Erdős himself.
- 7373737373 6mo agoThis has existed for a few months, but there aren't any reports of (unsuccessful) attempts: https://github.com/google-deepmind/formal-conjectures https://github.com/google-deepmind/formal-conjectures
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- LPisGood 6mo agoSome Erdős problems are basically trivial using sophisticated techniques that were developed later. I remember one of my professors, a coauthor of Erdős boasted to us after a quiz how proud he was that he was able to assign an Erdős problem that went unsolved for a while as just a quiz problem for his undergrads.
- vessenes 6mo agoTao mentions that the conventional approach for this problem seems to be a dead-end, but it’s apparently a super ‘obvious’ first step. This seems very hopeful to me — in that we now have a new approach line to evaluate / assess for related problems.
- CSMastermind 6mo agoWorth mentioning, though, that people have already tried running all of them through LLMs at this point. So this is proof of the models actually getting stronger (previous generations of LLMs were unable to solve this one).
- Tarq0n 6mo agoNot definitively. LLMs are stochastic with respect to input, temperature and the exact prompt. It's possible that the model was already capable of it but never received the exact right conditions to produce this output.
- winwang 6mo agoObviously nowhere near Erdos problem complexity but I've been using GPT (in Codex) to prove a couple theorems (for algos) and I've found it a bit better than Claude (Code) in this aspect.
- jzer0cool 6mo agoCould someone share a bit into the problem and the key portion from proof? For someone just knowing basics on proofs.
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- brcmthrowaway 6mo agoThis is not a good Saturday night for humanity
- echelon 6mo agoNow do P vs NP. If/when these things solve our hardest problems, that's going to lead to some very uncomfortable conversations and realizations.
- lucasgerads 6mo agoI feel like a year ago I would have said impossible. Now, I am not so sure anymore. Although, if I wrote the prompt and the correct result would be presented to me I wouldn't even know. Would still need a mathematician to verify it.
- ngruhn 6mo agoNah, people are going to say: It just used these 500 weird tricks from all kinds of different areas. A human could totally have done it. Nobody looked. I guess P/NP wasn't that hard after all.
- shybear 6mo agoIt seems like alot of scientific advancements occurred by someone applying technique X from one field to problem Y in another. I feel like LLMs are much better at making these types of connections than humans because they 1) know about many more theories/approaches than a single human can 2) don't need to worry about looking silly in front of their peers.
- bojo 6mo agoThis is what I have been doing. I don't think I've made any amazing breakthroughs, but at the same time I can't help but feel like I've come across some white paper-worthy realizations. Being able to correlate across a lot of domains I feel like I intuitively understand but have no depth of knowledge has been a fun exercise in LLM experimentation.
- freakynit 6mo agoThis is what I personally consider as "reasoning" ... knowledge generalization and application across domains.
- jdub 6mo agoLess reasoning than a dimension of brute force unfamiliar to human brains.
- worldsavior 6mo agoFamiliar but isn't effective enough for surviving.
- squidbeak 6mo agoTrying to diminish this as brute force (something by the way that is categorically not 'unfamiliar to human brains' - as anyone who has every worked on complex slippery problems will tell you) is foolish, when the models hypothesize along the way to their solutions. That's reasoning.
- 6mo ago
- nadermx 6mo agoThis just shows that with the right training, in this case a thesis on erdos problems, they where able to prompt and check the output. So still needed the know how to even being to figure it out. "Lichtman proved Erdős right as part of his doctoral thesis in 2022."
- fwipsy 6mo agoLichtman is an expert who commented for the story. Liam Price is the one who prompted ChatGPT. "He’s 23 years old and has no advanced mathematics training."
- nadermx 6mo ago“I didn’t know what the problem was—I was just doing Erdős problems as I do sometimes, giving them to the AI and seeing what it can come up with,” he says. “And it came up with what looked like a right solution.” "He sent it to his occasional collaborator Kevin Barreto, a second-year undergraduate in mathematics at the University of Cambridge." So basically two undergrads/graduates in math, "advanced" is subjective at that point.
- fwipsy 6mo agoI don't see where it says Price was an undergraduate/graduate in math.
- nadermx 6mo agoI don't see where it doesn't say he is, I feel its implied. Another source, proves me right? https://www.newscientist.com/article/2511954-amateur-mathematicians-solve-long-standing-maths-problems-with-ai/ https://www.newscientist.com/article/2511954-amateur-mathema... https://archive.is/oQvO4 https://archive.is/oQvO4
- fwipsy 6mo agoIt's implied by "no advanced mathematics training?" The article you linked (thanks for the unpaywalled link, by the way) describes him only as an amateur mathematician, but describes Barreto as a math student. If they were both math students, I feel it would say so? Or perhaps you're arguing it's implicit in him having solved the problem? If so, you're just assuming your conclusion. "AI didn't prove it by itself; Price was a mathematician. Well, he must have been a mathematician to be able to prove it!"
- CSMastermind 6mo agoFor the uninitiated, Paul Erdős was a pretty famous but very eccentric mathematician who lived for most of the 1900s. He had a habit of seeking out and documenting mathematical problems people were working on. The problems range in difficulty from "easy homework for a current undergrad in math" to "you're getting a Fields Medal if you can figure this out". There's nothing that really connects the problems other than the fact that one of the smartest people of the last 100 years didn't immediately know the answer when someone posed it to him. One of the things people have been doing with LLMs is to see if they can come up with proofs for these problems as a sort of benchmark. Each time there's a new model release a few more get solved.
- energy123 6mo ago> Each time there's a new model release a few more get solved. I'm no expert, but based on the commentary from mathematicians, this Erdős proof is a unique milestone because the problem received previous attention from multiple professional mathematicians, and the proof was surprising, elegant, and revealed some new connections. The previous ChatGPT Erdős proofs have been qualitatively less impressive, more akin to literature search or solving easier problems that have been neglected. Reading the prompt[1], one wonders if stoking the model to be unconventional is part of the success: "this ... may require non-trivial, creative and novel elements" [1] https://chatgpt.com/share/69dd1c83-b164-8385-bf2e-8533e9baba9c https://chatgpt.com/share/69dd1c83-b164-8385-bf2e-8533e9baba...
- sigmoid10 6mo ago>one wonders if stoking the model to be unconventional is part of the success I've long suspected that a lot of these model's real capabilities are still locked behind certain prompts, despite the big labs spending tons of effort on making default responses to simple prompts better. Even really dumb shit like "Answer this: ..." vs "Question: ..." vs "... you'll be judged by <competitor>" that should have zero impact in an ideal world can significantly impact benchmark results. The problem is that you can waste a ton of time finding the right prompt using these "dumb" approaches, while the model actually just required some very specific context that was obvious to you and not to it in many day-to-day situations. My go to method is still to have the model ask me questions as the very first step to any of these problems. They kind of tried that with deep research since the early o-series, but it still needs improvement.
- Rahil_Jain 6mo ago[dead]
- dataflow 6mo agoQuestion for those who believe LLMs aren't intelligent and are merely statistical word predictors: how do you reconcile such achievements with that point of view? (To be clear: I'm not agreeing or disagreeing. I sometimes feel the same too. I'm just curious how others reconcile these.)
- fc417fc802 6mo agoThose things aren't mutually exclusive. They are demonstrably statistical token predictors (go examine an open source implementation) and they clearly exhibit intelligence.
- downboots 6mo agoIt doesn't matter if you use a car or go there walking. If your goal is cave exploration, the tools are irrelevant.
- azan_ 6mo agoBut in this specific case AI actually explored the cave for you. Comparing it to car getting you to the cave is really bad comparison.
- utopiah 6mo agoMandatory disclaimers https://github.com/teorth/erdosproblems/wiki/Disclaimers-and-caveats https://github.com/teorth/erdosproblems/wiki/Disclaimers-and...
- logicprog 6mo agoThey explicitly say many of these disclaimers don't apply in the article.
- utopiah 6mo agoWhich one do you trust most, the disclaimers or the article?
- logicprog 6mo agoYou're not arguing in good faith here, but just to make this apparent to everyone else: the disclaimers talk about the general case of Erdos problems as a whole. The article explicitly acknowledges them, but then says that the disclaimers don't apply in this specific case: > ...experts have warned that these problems are an imperfect benchmark of artificial intelligence’s mathematical prowess. They range dramatically in both significance and difficulty, and many AI solutions have turned out to be less original than they appeared. The new solution—which Price got in response to a single prompt to GPT-5.4 Pro and posted on www.erdosproblems.com, a website devoted to the Erdős problems, just over a week ago—is different. The problem it solves has eluded some prominent minds, bestowing it some esteem. And more importantly, the AI seems to have used a totally new method for problems of this kind. It’s too soon to say with certainty, but this LLM-conceived connection may be useful for broader applications—something hard to find among recently touted AI triumphs in math. So I don't see why I have to trust only one of only the other. Furthermore, their assessment is backed up by direct quotes from Tao himself: > “This one is a bit different because people did look at it, and the humans that looked at it just collectively made a slight wrong turn at move one,” says Terence Tao, a mathematician at the University of California, Los Angeles, who has become a prominent scorekeeper for AI’s push into his field. “What’s beginning to emerge is that the problem was maybe easier than expected, and it was like there was some kind of mental block.”... “We have discovered a new way to think about large numbers and their anatomy,” Tao says. “It’s a nice achievement. I think the jury is still out on the long-term significance.”
- 3vo-ai 6mo ago[dead]
- yrds96 6mo agoGiven by the fact that the problem is 60 year old, isn't there a chance this was indirect solved already and the model just crossed informations to figure out the problem? By looking the website this problem was never discussed by humans. The last comments were about gpt discovering it. I was expecting older comments coming to a 60 year old problem. Am I missing something? Great discovery though, there might be problems like that same case that worth a try for a "gpt check"
- whiplash451 6mo agoTo some extent, does it matter? If models are able to pull and join information that already existed in pieces but humankind never discovered by itself, doesn’t this count towards progress anyways?
- fuglede_ 6mo agoIt would be very helpful to know in understanding the capabilities of the models; and in getting intuition about where they are best applicable. If the reason it was able to output the proof is that it happened to be included in an in-house university report written in Georgian, then that would make it less useful for research than if it's new entirely.
- traes 6mo agoExceedingly unlikely. This was one of the more discussed Erdos problems, and multiple experts have attested to the technique's novelty. If you're referring to the lack of comments on the erdosproblems website, that doesn't really mean much. From its own blog[0], the site was only started in 2023 and only really gained momentum as a place to discuss AI solving attempts, you aren't going to see serious mathematicians discussing the problems there even if there have been significant efforts to solve it. [0]: https://www.erdosproblems.com/forum/thread/blog:1 https://www.erdosproblems.com/forum/thread/blog:1
- yrds96 6mo agoYeah I was referring to the lack of comments on the website. Thanks! That answer a lot and makes everything more interesting.
- wiseowise 6mo agoWake me up when it creates cancer cure or fusion reactor.
- jchook 6mo agoIs the conjecture not trivially sound at an intuition level? It's surprising that this proof was difficult.
- booleandilemma 6mo agoWhat’s beginning to emerge is that the problem was maybe easier than expected, and it was like there was some kind of mental block Hindsight is 20/20.
- Aboutplants 6mo agomost likely true, the near value of AI will finding the low hanging fruit that has been missed. And hopefully those discoveries will prove valuable to current processes
- Drupon 6mo ago>ChatGPT, prompted by an amateur, solves an Erdős problem. There, fixed that for you.
- quijoteuniv 6mo agoAI is my favourite weird collaborator
- gorgoiler 6mo agoI asked ChatGPT to draw the outline of an ellipse using Unicode braille. I asked for 30x8 and it absolutely nailed it. A beautiful piece of ascii (er, Unicode) art. But I wanted to mark the origin! So I asked for a 31x7 ellipse instead. It completely flubbed it, and for 31x9 too. When a model gives a really good answer, does that just mean it’s seen the problem before? When it gives a crappy answer, is that not simply indicating the problem is novel?
- ghusbands 6mo agoDo you posit that there are enough examples of 30x8 ellipses encoded in braille online for ChatGPT to learn from but not 31x7 or 31x9 ellipses? That seems unlikely.
- gorgoiler 6mo agoYes, or the model got lucky with the quality of output for a particular combination of my prompt and the reasoning behind its answer that lined up with something it had seen before — quality which it was unable to recreate under slightly different circumstances.
- Anon1096 6mo agoI wouldn't ask an LLM to output this directly. For an ellipse ascii I would guess that having it write a python program to generate it and then run it would work much better. Using claude sonnet 4.6 on a free account it seemed to work (sorry in advance if the hacker news formatting is horrendous) ⠀⠀⠀⠀⠀⣀⣠⠤⠔⠒⠒⠋⠉⠉⠉⠉⠉⠉⠉⠙⠒⠒⠢⠤⣄⣀⠀⠀⠀⠀⠀ ⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀⠀ ⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆ ⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸ ⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏ ⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀ ⠀⠀⠀⠀⠀⠉⠙⠒⠢⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠔⠒⠋⠉⠀⠀⠀⠀⠀
- gus_massa 6mo agoYou can use two spaces at the beginning of each line to trigger the "code" mode. I tried to reconstruct your drawing, but perhaps I didn't guess correctly: ⠀⠀⠀⠀⠀⣀⣠⠤⠔⠒⠒⠋⠉⠉⠉⠉⠉⠉⠉⠙⠒⠒⠢⠤⣄⣀ ⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀ ⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆ ⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸ ⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏ ⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁ ⠀⠀ ⠀⠀⠉⠙⠒⠢⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠔⠒⠋⠉⠀⠀⠀⠀⠀ Edit: I had to delete the two first spaces or each line and replace them with newly typed spaces from my keyboard. Perhaps there is some white-space-unicode-magic-character that is confusing HN.
- ccppurcell 6mo agoI will get downvoted for this but I can't help thinking that billions of dollars have gone into chatgpt over a period of years and an LLM can direct all its "attention" (in a metaphorical sense) on one problem. I think if you gave top mathematicians a few million (so a fraction of a percent of chatgpt budget) to solve this problem over four years, they probably would have at least made significant progress. I don't think chatgpt has solved thousands of similar problems (even stretching that across all ham disciplines). Basically my thesis is that universal basic income could have had a similar impact, and also encouraged human flourishing elsewhere.
- deleted 6mo ago[deleted]
- coalstartprob 6mo ago[dead]
- notahacker 6mo agoThere are literally millions of people who receive incomes from states which don't restrict them from spending 90% of their waking hours studying mathematics proofs, if that is what they wanted to do. Most of them do not and overwhelmingly could not, even if we took the opposite tack and made their welfare or pensions or even university fees contingent upon them solving mathematics problems. Topping up the global welfare budget by a couple of hundred billion might meaningfully improve some people's lives, but even with the most sceptical take on AI usefulness it's hard to imagine it producing more research than went into and came out of ChatGPT.... We also actually do devote millions in public funds to enable top mathematicians to spend much of their time studying mathematical problems, but it turns out that there are a lot of problems, solving them is hard, and sometimes they like to spend their time devising new problems instead. Perhaps some people currently dedicating their efforts to writing trading algorithms would also prove adept at devising novel proofs to more abstract mathematics problems, but I don't think UBI is changing their personal priorities...
- traes 6mo agoDiscussed at the time: https://news.ycombinator.com/item?id=47774494 https://news.ycombinator.com/item?id=47774494
- nomilk 6mo agoA similar announcement was made a few months ago, and Terence Tao came out a few days later and said it wasn't what it seemed at first, in that it was a rediscovery of an already known (albeit esoteric) result...
- logicprog 6mo agoThey literally have a quote from Tao in the article saying it was a novel approach humans hadn't tried, and that the problem hadn't been solved even after a lot of professional attention.
- cubefox 6mo agoCurrent headline: "An amateur just solved a 60-year-old math problem—by asking AI" A more honest title would be: "An AI just solved a 60-year-old math problem—after being asked by amateur" (Imagine the headline claimed instead that a professor just solved a math problem by asking a grad student.)
- ngruhn 6mo agoPrevious problems solved by AI had some amount of expert guidance/steering. Here, I guess the emphasis is that there was none of that.
- tokenhub_dev 6mo ago[dead]
- openclawclub 6mo ago[flagged]
- mrabcx 6mo agoCan the other AI agents such as Gemini, Calude or Deepseek etc also solve this problem?
- mfgadv99 6mo ago[flagged]
- dnnddidiej 6mo agoHow do you get real mathematicians to check the potential slop. At some point there will be spam to Tao from claws finding problens to solve and submitting maybe proofs/answers.
- brohee 6mo agoIn the end "proofs" that are not machine checked will be left unread unless submitted by someone very respected in the field...
- etaKl 6mo ago1) How do you know the clanker respects the instruction not to search the internet? 2) Jared Lichtman is indeed a mathematician at Stanford University but involved in the AI startup math.inc, which seems more relevant here. Terence Tao is involved in a partership program with that startup. 3) Liam Price is a general AI booster on Twitter. A lot of AI boosting on Twitter is not organic and who knows what help he got. Nothing in this Twitter is organic. 4) Scientific American is owned by Springer Nature, which is an AI booster: https://group.springernature.com/gp/group/ai https://group.springernature.com/gp/group/ai
- lima 6mo ago> How do you know the clanker respects the instruction not to search the internet? You can't, but given that it's a previously unsolved problem, it doesn't seem relevant? (nor are the author's potential biases - the claims are easily verified independently)
- lakkv 6mo agoThe fact that disclosures that would have been standard in 2000 are now downranked to limit their reach shows that AI discussion is indistinguishable from doubting the Archangel Moroni on an LDS forum. Maybe that isn't fair, probably the LDS people are more open minded that the pro-AI people.
- anthonyrstevens 6mo agoThe parent comment you refer to is part disclosure listing, part bad-faith conspiracy blatting
- laurentiurad 6mo agoThis program was brought to you by the private equity engagement pod.
- meken 6mo ago> “What’s beginning to emerge is that the problem was maybe easier than expected, and it was like there was some kind of mental block.” Even if AI never progresses past this point, it still seems like a huge win for math research to “clear the deck” of these.
- deleted 6mo ago[deleted]
- wslh 6mo agoThe current state of AI is incredible, and useful and doesn't need to reach AGI to be revolutionary. For example, I uploaded a conversation between a few people and not only asked about translating the text but doing a psychological analysis on turn-turning and other conversational cues. Just around a decade ago, the speech-to-text Dragon Naturally Speaking[1] was not reliable with only one speaker without any background noise. [1] https://en.wikipedia.org/wiki/Dragon_NaturallySpeaking https://en.wikipedia.org/wiki/Dragon_NaturallySpeaking
- mannanj 6mo agoDo we get the information necessary for this solutions if the model providers are improvising or hiding or changing the thinking for security/IP purposes?
- nekusar 6mo agoIf anything, this shows that by shoving all the knowledge we have currently in a blender, that we've actually solved a LOT more than we think. This LLM prompt didnt create *new* proofs. It used existing human knowledge from other areas that arent well shared, and connected associations to the problem at hand. It was already mostly solved. The LLM just basically did the usual pattern matching of jigsaw pieces and connected the 2 domains together. We see that with "The LLM took an entirely different route, using a formula that was well known in related parts of math, but which no one had thought to apply to this type of question." in the article. There's still a TON of stuff that can be done to connect domains together. And that alone is amazingly powerful. But humans are still doing the creative work at the edges. These stochastic word-calculus machines are not yet able to generate new thought, or process absolutely current research. It'll probably get there... but we'll likely need thinking machines. Thats also the hell scenario too.
- contubernio 6mo agoThat ai can help solve a problem perhaps indicates that the problem is shallow.
- cm2012 6mo agoNo true Scotsman fallacy
- lqstuart 6mo agoBuried pretty deep in the article > “The raw output of ChatGPT’s proof was actually quite poor. So it required an expert to kind of sift through and actually understand what it was trying to say,” Lichtman says. But now he and Tao have shortened the proof so that it better distills the LLM’s key insight. I guess “ChatGPT came up with a novel approach to a problem that later turned out not to be totally stupid and terrible for once” isn’t as catchy of a headline
- FrustratedMonky 6mo agoHow often are humans initial key insights, also sublimely distilled and beautiful. This is like comparing someone's first draft, with a final published paper.
- arcticfox 6mo agoThat should be buried, I agree 100% with their headline and structure over yours. For comparison, if the amateur did it by hand but the result was sloppy to read, would you prefer "Amateur solves an Erdos problem" or "Amateur came up with a novel approach to a problem that later turned out not to be totally stupid and terrible for once"?
- deleted 6mo ago[deleted]
- vovavili 6mo agoI wouldn't expect a hand-crafted proof by an amateur to be much different.
- stingraycharles 6mo agoWouldn’t the expectation for ChatGPT be that it presents a well refined report, rather than hand crafted proof / notes? Because from what I gather, they basically had to go through the equivalent of a pile of notes to find the crux.
- cccbbbaaa 6mo ago
- IAmGraydon 6mo agoThe emotional/defensive reactions I’m seeing here are telling. This is an interesting result, to say the least, as it appears to be the first solving of an Erdös problem completely unassisted. Let’s give it some time to make sure no other information comes to light.
- code51 6mo agoWhy on earth is nobody here talking about the sudden jump to use von Mangoldt function? The reasoning trace never types Λ, never types "von Mangoldt", and never invokes ∑_{q|n} Λ(q) = log n. There is a clear discontinuity at play. I remember an article on this, maybe a comment by Terence Tao himself, seen here, but cannot find it.
- sweezyjeezy 6mo agoI think that the thought trace is definitely incomplete - you can see cases where it is like and "let's calculate the integral:[no integral calculated]". The train of thought it's on towards the end of the trace looks like an entirely different approach than what it ends up returning, so I think we are just not seeing the part where it hits on the right approach (sadly).
- culi 6mo agoDoes DeepSeek's solution look more traceable? https://chat.deepseek.com/share/nyuz0vvy2unfbb97fv https://chat.deepseek.com/share/nyuz0vvy2unfbb97fv
- pelorat 6mo agoThought traces are indeed not an accurate representation of what models actually do. If you ask an AI model to add two values it will do so, then in the next prompt ask it to explain the algorithm it used, it will regurgitate that it used some standard textbook method, whilst in reality it used a completely different algorithm. Thinking LLMs don't record the neural pathways they used.
- dataviz1000 6mo agoDuring training they gate with a lot of guardrails the format of the reasoning tokens output. They don't just use a reward for getting the correct answer during training but also reward human readable output. That said, if they didn't, the reasoning tokens that are the most efficient to get to the final correct answer during training would most likely look like a lot of gibberish. There is a relationship between the tokens in the output in the model's vector space, that is the most important, and something hidden we will never see.
- JonChesterfield 6mo agoYou too can solve maths problems by: 1. Generating enormous amounts of text 2. Persuading a mathematician to look closely at it 3. Announcing success if they conclude it is a proof This is deeply disappointing relative to "chatgpt found a proof that isabelle verifies" or similar, especially the part where a mathematician spends (presumably hours) reading through the llm output.
- booleandilemma 6mo agoI think large proofs done by humans also require hours of verification by other mathematicians, checking for "bugs" in a sense. I don't think they're obviously correct, I think it's like more like doing a code review.
- iwontberude 6mo agoKey quote I went into the article looking for and was not disappointed “The raw output of ChatGPT’s proof was actually quite poor. So it required an expert to kind of sift through and actually understand what it was trying to say,” Lichtman says.
- mettamage 6mo agoSo when will the Riemann hypothesis be proven or disproven?
- crsn 6mo agoThe headline misses the most impressive part: ChatGPT one-shotted the problem. No turns, no retries, no mid-thinking steering from the user. One-shotting a problem like this would have been nearly unthinkable in 2025.
- Aboutplants 6mo agoThis was my main takeaway, it didn’t need the type of guidance we are accustomed to. A peak into the future perhaps? At least the future they are striving for
- Grappelli 6mo ago[dead]
- Pixelora 6mo agoInteresting perspective. I think simplicity in products is often underrated.
- atlasagentsuite 6mo ago[dead]
- gxt 6mo agoAnybody with access to these models can challenge it to test the hypothesis that spacetime is a 4d viscous fluid with the speed of light being spacetime's sound barrier, mass relating to viscosity, blackholes being cavitation bubbles, Hawking radiation our perception of surface tension, and gravity just being a pressure differencial? Thanks