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Erdos 281 solved with ChatGPT 5.2 Pro

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Re: Erdos 281 solved with ChatGPT 5.2 Pro

#161

Post by Davidzheng »

xigoi wrote: post #160
Davidzheng wrote: post #135

Earlier quoted context omitted.

I actually don't think the reason is that they are easier than other open math problems. I think it's more that they are "elementary" in the sense that the problems usually don't require a huge amount of domain knowledge to state.

The Collatz conjecture can be stated using basic arithmetic, yet LLMs have not been able to solve it.

I agree it's easier than Collatz. I just mean I am not sure it's much easier than many currently open questions which are less famous but need more machinery.

Re: Erdos 281 solved with ChatGPT 5.2 Pro

#162

Post by Eufrat »

There was a post about Erdős 728 being solved with Harmonic’s Aristotle a little over a week ago [1] and that seemed like a good example of using state-of-the-art AI tech to help increase velocity in this space.

I’m not sure what this proves. I dumped a question into ChatGPT 5.2 and it produced a correct response after almost an hour [2]?

Okay? Is it repeatable? Why did it come up with this solution? How did it come up with the connections in its reasoning? I get that it looks correct and Tao’s approval definitely lends credibility that it is a valid solution, but what exactly is it that we’ve established here? That the corpus that ChatGPT 5.2 was trained on is better tuned for pure math?

I’m just confused what one is supposed to take away from this.

[1] https://news.ycombinator.com/item?id=46560445

[2] https://chatgpt.com/share/696ac45b-70d8-8003-9ca4-320151e081...

Re: Erdos 281 solved with ChatGPT 5.2 Pro

#163

Post by fatherwavelet »

kristopolous wrote: post #125
energy123 wrote: post #113

A surprising % of these LLM proofs are coming from amateurs. One wonders if some professional mathematicians are instead choosing to publish LLM proofs without attribution for career purposes.

It's probably from the perennial observation "This LLM is kinda dumb in the thing I'm an expert in"

This is just not true at this point but believe whatever you want to believe.

Re: Erdos 281 solved with ChatGPT 5.2 Pro

#164

Post by Paracompact »

Davidzheng wrote: post #127
sequin wrote: post #10

FWIW, I just gave Deepseek the same prompt and it solved it too (much faster than the 41m of ChatGPT). I then gave both proofs to Opus and it confirmed their equivalence. The answer is yes. Assume, for the sake of contradiction, that there exists an \(\epsilon > 0\) such that for every \(k\), there exists a choice of congruence classes \(a_1^{(k)}, \dots, a_k^{(k)}\) for which the set of integers not covered by the f…

"Since \(U_{k+1} \subseteq U_k\), the sets \(U_k\) are decreasing and periodic, and their intersection \(U = \bigcap_{k \ge 1} U_k\) has density \(d = \lim_{k \to \infty} d_k \ge \epsilon\)." Is this enough? Let $U_k$ be the set of integers such that their remainder mod 6^n is greater or equal to 2^n for all 1<n<k. Density of each $U_k$ is more than 1/2 I think but not the intersection (empty) right?

Indeed. Your sets are decreasing periodic of density always greater than the product from k=1 to infinity of (1-(1/3)^k), which is about 0.56, yet their intersection is null.

This would all be a fairly trivial exercise in diagonalization if such a lemma as implied by Deepseek existed.

(Edit: The bounding I suggested may not be precise at each level, but it is asymptotically the limit of the sequence of densities, so up to some epsilon it demonstrates the desired counterexample.)

Re: Erdos 281 solved with ChatGPT 5.2 Pro

#166

Post by pelorat »

sekai wrote: post #153
mikert89 wrote: post #7

I have 15 years of software engineering experience across some top companies. I truly believe that ai will far surpass human beings at coding, and more broadly logic work. We are very close

> I have 15 years of software engineering experience across some top companies. I truly believe that ai will far surpass human beings at coding, and more broadly logic work. We are very close Coding was never the hard part of software development.

Getting the architecture mostly right, so it's easy to maintain and modify in the future is IMO hard part, but I find that this is where AI shines. I have 20 years of SWE experience (professional) and (10 hobby) and most of my AI use is for architecture and scaffolding first, code second.

Re: Erdos 281 solved with ChatGPT 5.2 Pro

#167

Post by fatata123 »

fatherwavelet wrote: post #163
kristopolous wrote: post #125

Earlier quoted context omitted.

It's probably from the perennial observation "This LLM is kinda dumb in the thing I'm an expert in"

This is just not true at this point but believe whatever you want to believe.

[dead]

Re: Erdos 281 solved with ChatGPT 5.2 Pro

#169

Post by danielbln »

glemion43 wrote: post #158
cubefox wrote: post #117

Earlier quoted context omitted.

This illustrates how unimportant this problem is. A prior solution did exist, but apparently nobody knew because people didn't really care about it. If progress can be had by simply searching for old solutions in the literature, then that's good evidence the supposed progress is imaginary. And this is not the first time this has happened with an Erdős problem. A lot of pure mathematics seems to consist in solving nea…

It shows that a 'llm' can now work on issues like this today and tomorrow it can do even more. Don't be so ignorant. A few years ago NO ONE could have come up with something so generic as an LLM which will help you to solve this kind of problems and also create text adventures and java code.

The goal posts are strapped to skateboards these days, and the WD40 is applied to the wheels generously.

Re: Erdos 281 solved with ChatGPT 5.2 Pro

#170

Post by TheOtherHobbes »

cubefox wrote: post #159
baq wrote: post #155

Earlier quoted context omitted.

You’re confusing immediately useful with eventually useful. Pure maths has found very practical applications over the millennia - unless you don’t consider it pure anymore, at which point you’re just moving goalposts.

No, I'm not confusing that. Read the linked comment if you're interested.

You are confusing that. The biggest advancements in science are the result of the application of leading-edge pure math concepts to physical problems. Netwonian physics, relativistic physics, quantum field theory, Boolean computing, Turing notions of devices for computability, elliptic-curve cryptography, and electromagnetic theory all derived from the practical application of what was originally abstract math play.

Among others.

Of course you never know which math concept will turn out to be physically useful, but clearly enough do that it's worth buying conceptual lottery tickets with the rest.

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