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.
Yes, 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.
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.
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."
> Every Mathematician Has Only a Few Tricks
>
> A long time ago an older and well-known number theorist made some disparaging remarks about Paul Erdös’s work.
> You admire Erdös’s contributions to mathematics as much as I do,
> and I felt annoyed when the older mathematician flatly and definitively stated
> that all of Erdös’s work could be “reduced” to a few tricks which Erdös repeatedly relied on in his proofs.
> What the number theorist did not realize is that other mathematicians, even the very best,
> also rely on a few tricks which they use over and over.
> Take Hilbert. The second volume of Hilbert’s collected papers contains Hilbert’s papers in invariant theory.
> I have made a point of reading some of these papers with care.
> It is sad to note that some of Hilbert’s beautiful results have been completely forgotten.
> But on reading the proofs of Hilbert’s striking and deep theorems in invariant theory,
> it was surprising to verify that Hilbert’s proofs relied on the same few tricks.
> Even Hilbert had only a few tricks!
>
> - Gian-Carlo Rota - "Ten Lessons I Wish I Had Been Taught"
The combinatorial nature of trying things randomly means that it would take millennia or longer for light-speed monkeys typing at a keyboard, or GPUs, to solve such a problem without direction.
By now, people should stop dismissing RL-trained reasoning LLMs as stupid, aimless text predictors. They wouldn’t say the same thing to high-achieving, but non-creative, college students who can only solve hard conventional problems.
Yes, current LLMs likely still lack some major aspects of intelligence. They wouldn’t be able to come up with general relativity on their own with only training data up to 1905.
Neither did the vast majority of physicists back then.
Wait, what do you mean "LLMs are still absolutely useless at actual maths computation"? I rely on them constantly for maths (linear algebra, multivariable calc, stat) --- literally thousands of problems run through GPT5 over the last 12 months, and to my recollection zero failures. But maybe you're thinking of something more specific?
They are bad at math. But they are good at writing code and as an optimization some providers have it secretly write code to answer the problem, run it and give you the answer without telling you what it did in the middle part.
What tier are you using? I have run lots of problems and am very impressed, but I find stupid errors a lot more frequently than that, e.g., arithmetic errors buried in a derivation or a bad definition, say 1/15 times. I would love to get zero failures out of thousands of (what sounds like college-level math) posed problems.
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.
And how about the creative rationalizations about how statistical text generation is actual intelligence? As if there is any intent or motive behind the words that are generated or the ability to learn literally any new thing after it has been trained on human output?
For 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.
What 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?
We 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.
There'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.
I think one day the VCs will have given the monkeys on typewriters enough money that these kinds of comments can be generated without human intervention.
> “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.
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.
My 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.
I've tried my hand at a few of the Erdos problems and came up short, you didn't hear about them. But if a Mathematician at Harvard solved on, you would probably still hear about it a bit. Just the possibility that a pro subscription for 80 minutes solved an Erdos problem is astounding. Maybe we get some researchers to get a grant and burn a couple data centers worth of tokens for a day/week/month and see what it comes up with?
I 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.
Also, it's one thing if the AI age means we all have to adopt to using AI as a tool, another thing entirely if it means the only people who can do useful research are the ones with huge budgets.
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.
Maybe... 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...
> “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.
Scientific 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.
Here is the chat:
Then "Thought for 80m 17s"https://chatgpt.com/share/69dd1c83-b164-8385-bf2e-8533e9baba...
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.
Tried the same prompt and ended up no where close on the free plan.
Is there a known lag that it takes the Pro plan's abilities to migrate to the free plans?
GPT 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.
Yes, 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.
You should pay for it if you find value in it.
Tangential 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.
Do not use the free plan. It is not good.
Does the free plan even have access to thinking models?
Technically yes, gpt-5.4-mini is available on the free plan
Was this a surprise?
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.
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."
> "a broken clock is right twice a day."
The combinatorial nature of trying things randomly means that it would take millennia or longer for light-speed monkeys typing at a keyboard, or GPUs, to solve such a problem without direction.
By now, people should stop dismissing RL-trained reasoning LLMs as stupid, aimless text predictors. They wouldn’t say the same thing to high-achieving, but non-creative, college students who can only solve hard conventional problems.
Yes, current LLMs likely still lack some major aspects of intelligence. They wouldn’t be able to come up with general relativity on their own with only training data up to 1905.
Neither did the vast majority of physicists back then.
Yeah, they're great at interpolation - they'll just never be worth much at extrapolation.
Luckily for us, whole fortunes can be made by filling in the blanks between what we know and what we realize.
Wait, what do you mean "LLMs are still absolutely useless at actual maths computation"? I rely on them constantly for maths (linear algebra, multivariable calc, stat) --- literally thousands of problems run through GPT5 over the last 12 months, and to my recollection zero failures. But maybe you're thinking of something more specific?
They are bad at math. But they are good at writing code and as an optimization some providers have it secretly write code to answer the problem, run it and give you the answer without telling you what it did in the middle part.
Someone should tell the mathematicians if they use a calculator or a whiteboard or heavens forbid a computer they are "bad at math".
Are they bad at math? Or are they bad at arithmetic?
if you don't know much math, it's easy to confuse the two
What tier are you using? I have run lots of problems and am very impressed, but I find stupid errors a lot more frequently than that, e.g., arithmetic errors buried in a derivation or a bad definition, say 1/15 times. I would love to get zero failures out of thousands of (what sounds like college-level math) posed problems.
The ultimate generalist
Also just the sheer value of brute force.
80 hours! 80 hours of just trying shit!
It's 80 minutes, not 80 hours.
and you can be sure mathematicians spent way more than 80 hrs on it
80 minutes! 80 minutes of just trying shit!
... shit that solved an apparently significant Erdős problem.
That is not nothing, no matter how much you hate AI.
It shows that AI is apparently very good at brute-forcing.
This isn't brute force.
How long do you figure it’d take to solve the problem yourself?
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.
There 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
that's actually a brilliant idea
Tried w/ 5.5 Pro. 17 minutes:
https://chatgpt.com/share/69ed8e24-15e8-83ea-96ac-784801e4a6...
I wonder if the rationalizations people come up with for why this isn't real intelligence will be as creative as ChatGPTs solution.
Remember when people thought multiplying numbers, remembering a large number of facts, and being good at rote calculations was intelligence?
Some people think that multiplying numbers, remembering a large number of facts, and being good at calculations is intelligence.
Most intelligent people do not think that.
Eventually, we will arrive at the same conclusion for what LLMs are doing now.
You're really telling on yourself if you think LLM is intelligence
And how about the creative rationalizations about how statistical text generation is actual intelligence? As if there is any intent or motive behind the words that are generated or the ability to learn literally any new thing after it has been trained on human output?
This is real intelligence is the bear position, so I think it’s real intelligence.
For 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.
There's humans that have memory issues, or full blown Anterograde amnesia.
There are humans who can’t read. That doesn’t mean Grammarly is “intelligent”. These things are tools - nothing more, nothing less.
All modern harnesses write memory files for context later.
What 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?
We 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.
There'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.
You are confusing lack of intelligence with the presence of impairment.
I think one day the VCs will have given the monkeys on typewriters enough money that these kinds of comments can be generated without human intervention.
> “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.
https://archive.ph/2w4fi
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.
referring to Tao as just a 'mathematician' gave me a good chuckle
Big if true.
WTF!?
My 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.
I've tried my hand at a few of the Erdos problems and came up short, you didn't hear about them. But if a Mathematician at Harvard solved on, you would probably still hear about it a bit. Just the possibility that a pro subscription for 80 minutes solved an Erdos problem is astounding. Maybe we get some researchers to get a grant and burn a couple data centers worth of tokens for a day/week/month and see what it comes up with?
Why do you care about either of those questions?
Because it could be a massive waste of time and money.
I 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.
Also, it's one thing if the AI age means we all have to adopt to using AI as a tool, another thing entirely if it means the only people who can do useful research are the ones with huge budgets.
Your logic undoes your point, because the kid who "solved" this technically didn't even have to invest in a degree.
America should fund tertiary education better, and that would solve even more problems.
Neither does the Collatz conjecture, Fermat's last theorem, ....
(Of course, those problems are on another plane than this one.)
But 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.
Maybe... 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...
Is it worth it to buy a super-yacht?
No.
I would question at $60k. At $100k is a steal.
Can you imagine how many bags of chips we could buy if we stopped funding cancer research?
It's so expensive!
Can you imagine how much ChatGPT cancer research we could fund if we stopped funding cancer research?
> 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?
> “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.
my guess would be due to having an interest in the field
Scientific 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.