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Amateur armed with ChatGPT solves an Erdős problem

scientificamerican.com

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Re: Amateur armed with ChatGPT solves an Erdős problem

#91

Post by resident423 »

thesmtsolver2 wrote: post #75
resident423 wrote: post #8

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.

Remember when people thought solving Erdos problems required intelligence? Is there anything an LLM could ever do that would cound as intelligence? Surely the trend has to break at some point, if so what would be the thing that crosses the line to into real intelligence?

Re: Amateur armed with ChatGPT solves an Erdős problem

#92

Post by nycdatasci »

adamgordonbell wrote: post #3

Here 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…

Tried 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-784801e4a6...

Re: Amateur armed with ChatGPT solves an Erdős problem

#93

Post by peteforde »

tomlockwood wrote: post #70
peteforde wrote: post #60

Earlier quoted context omitted.

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.

Getting off-topic, but as a successful high-school dropout I am compelled to remind anyone reading this that [the American] college [system] is a scam.

That's not to say that there aren't benefits to tertiary education, for many people in different contexts. It's just not the golden path that it's made out to be.

Many people currently in college are just wasting their money and should enroll in trades programs instead.

Meanwhile, nothing about being in or out of school is mutually exclusive to using LLMs as a force multiplier for learning - or solving math problems, apparently.

Re: Amateur armed with ChatGPT solves an Erdős problem

#94

Post by 0xBA5ED »

resident423 wrote: post #87
0xBA5ED wrote: post #55

Earlier quoted context omitted.

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?

Solving open math problems is strong evidence of intelligence so there's not really any need for rationalization? I don't understand why intelligence would require intent or motive? Isn't intent just the behaviour of making a specific thing happen rather than other things?

The LLM did not solve the problem.

Re: Amateur armed with ChatGPT solves an Erdős problem

#95

Post by vessenes »

jasonfarnon wrote: post #56
brianjking wrote: post #27

Earlier quoted context omitted.

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.

I 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.

Re: Amateur armed with ChatGPT solves an Erdős problem

#96

Post by tptacek »

schneems wrote: post #44
tptacek wrote: post #37

Earlier quoted context omitted.

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 would I do to demonstrate that they are bad at math? If by "maths" we mean things like working out a double integral for a joint probability problem, or anything simpler than that, GPT5 has been flawless.

Re: Amateur armed with ChatGPT solves an Erdős problem

#97

Post by vessenes »

abdullahkhalids wrote: post #86
ripped_britches wrote: post #66

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

…and this problem was in fact sourced directly from that list!

Re: Amateur armed with ChatGPT solves an Erdős problem

#98

Post by tptacek »

jasonfarnon wrote: post #65
tptacek wrote: post #37

Earlier quoted context omitted.

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?

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.

I have a standard OpenAI/ChatGPT Pro account; GPT5 is my daily driver for math, and Claude for code.

Re: Amateur armed with ChatGPT solves an Erdős problem

#99

Post by dinkumthinkum »

Eufrat wrote: post #11
gdhkgdhkvff wrote: post #9

Earlier quoted context omitted.

Why do you care about either of those questions?

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.

No meaningful, practical applications? You realize that sounds incredibly naive in the history of mathematics, right? People thought this way about number theory in general, and many other things that turned out to have quite important practical applications. Your statement is also a bit odd in that researchers are already paid throughout their whole careers to solve such problems. I don't know.

Re: Amateur armed with ChatGPT solves an Erdős problem

#100

Post by vessenes »

LPisGood wrote: post #88

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.

Tao 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.
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