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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.
https://chatgpt.com/share/69dd1c83-b164-8385-bf2e-8533e9baba...
What 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.
Tried w/ 5.5 Pro, Extended Thinking. 17 minutes:
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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-784801e4a6...
I 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.
I don't haven ChatGPT but Gemini and Claude. But how do you make a language model think for 80 minutes ???
Mine took 20min. Pro.
https://chatgpt.com/share/69ed83b1-3704-8322-bcf2-322aa85d7a...
But I wish I was math smart to know if it worked or not.
>don't search the internet.
I think this was key. Otherwise the LLM could think it can't be done.
>>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-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
Another one for my theory that web search makes LLMs useless for anything other than searching the web.
> "Thought for 80m 17s"
Is there any good rule of thumb for how many kWh of electricity this is?
For 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.
It 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.
Buried 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
Some 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.
> “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.
> “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.
Why 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.
At 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.
I 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?
Humans 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.
The 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.
If 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.
Mandatory disclaimers https://github.com/teorth/erdosproblems/wiki/Disclaimers-and...
Given 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"
Discussed at the time: https://news.ycombinator.com/item?id=47774494
Current 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.)
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
Hindsight is 20/20.
1) 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:
Could someone share a bit into the problem and the key portion from proof? For someone just knowing basics on proofs.
The 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.
referring to Tao as just a 'mathematician' gave me a good chuckle
Anybody 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
I 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.
A 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...
Obviously 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.
This program was brought to you by the private equity engagement pod.
Do we get the information necessary for this solutions if the model providers are improvising or hiding or changing the thinking for security/IP purposes?
I wonder if the rationalizations people come up with for why this isn't real intelligence will be as creative as ChatGPTs solution.
Key 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.
Interesting perspective. I think simplicity in products is often underrated.
Can the other AI agents such as Gemini, Calude or Deepseek etc also solve this problem?
That ai can help solve a problem perhaps indicates that the problem is shallow.
So when will the Riemann hypothesis be proven or disproven?
Question 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.)
Now do P vs NP.
If/when these things solve our hardest problems, that's going to lead to some very uncomfortable conversations and realizations.
How 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.