Amateur armed with ChatGPT solves an Erdős problem

    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...

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.

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.

This is how I feel when I read any mathematics paper.

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?

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

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.

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"

"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.)

Hindsight is 20/20.

(To be clear: I'm not agreeing or disagreeing. I sometimes feel the same too. I'm just curious how others reconcile these.)

If/when these things solve our hardest problems, that's going to lead to some very uncomfortable conversations and realizations.

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."

There, fixed that for you.

Then my second question is how much VC money did all those tokens cost.

How is he even posing the question and having even a vague idea of what the proof means or how to understand it?