This is no longer true, a prior solution has just been found[1], so the LLM proof has been moved to the Section 2 of Terence Tao's wiki[2].

[1] - https://www.erdosproblems.com/forum/thread/281#post-3325

[2] - https://github.com/teorth/erdosproblems/wiki/AI-contribution...

EDIT: After reading a link someone else posted to Terrance Tao's wiki page, he has a paragraph that somewhat answers this question:

> Erdős problems vary widely in difficulty (by several orders of magnitude), with a core of very interesting, but extremely difficult problems at one end of the spectrum, and a "long tail" of under-explored problems at the other, many of which are "low hanging fruit" that are very suitable for being attacked by current AI tools. Unfortunately, it is hard to tell in advance which category a given problem falls into, short of an expert literature review. (However, if an Erdős problem is only stated once in the literature, and there is scant record of any followup work on the problem, this suggests that the problem may be of the second category.)

from here: https://github.com/teorth/erdosproblems/wiki/AI-contribution...

"Very nice! ... actually the thing that impresses me more than the proof method is the avoidance of errors, such as making mistakes with interchanges of limits or quantifiers (which is the main pitfall to avoid here). Previous generations of LLMs would almost certainly have fumbled these delicate issues.

...

I am going ahead and placing this result on the wiki as a Section 1 result (perhaps the most unambiguous instance of such, to date)"

The pace of change in math is going to be something to watch closely. Many minor theorems will fall. Next major milestone: Can LLMs generate useful abstractions?

Point in case: I just wanted to give z.ai a try and buy some credits. I used Firefox with uBlock and the payment didn't go through. I tried again with Chrome and no adblock, but now there is an error: "Payment Failed: p.confirmCardPayment is not a function." The irony is, that this is certainly vibe-coded with z.ai which tries to sell me how good they are but then not being able to conclude the sale.

And we will get lots more of this in the future. LLMs are a fantastic new technology, but even more fantastically over-hyped.

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 first \(k\) congruences has density at least \(\epsilon\).

For each \(k\), let \(F_k\) be the set of all infinite sequences of residues \((a_i)_{i=1}^\infty\) such that the uncovered set from the first \(k\) congruences has density at least \(\epsilon\). Each \(F_k\) is nonempty (by assumption) and closed in the product topology (since it depends only on the first \(k\) coordinates). Moreover, \(F_{k+1} \subseteq F_k\) because adding a congruence can only reduce the uncovered set. By the compactness of the product of finite sets, \(\bigcap_{k \ge 1} F_k\) is nonempty.

Choose an infinite sequence \((a_i) \in \bigcap_{k \ge 1} F_k\). For this sequence, let \(U_k\) be the set of integers not covered by the first \(k\) congruences, and let \(d_k\) be the density of \(U_k\). Then \(d_k \ge \epsilon\) for all \(k\). 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 \(d = \lim_{k \to \infty} d_k \ge \epsilon\). However, by hypothesis, for any choice of residues, the uncovered set has density \(0\), a contradiction.

Therefore, for every \(\epsilon > 0\), there exists a \(k\) such that for every choice of congruence classes \(a_i\), the density of integers not covered by the first \(k\) congruences is less than \(\epsilon\).

\boxed{\text{Yes}}

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...

I've "solved" many math problems with LLMs, with LLMs giving full confidence in subtly or significantly incorrect solutions.

I'm very curious here. The Open AI memory orders and claims about capacity limits restricting access to better models are interesting too.

One wonders if some professional mathematicians are instead choosing to publish LLM proofs without attribution for career purposes.

I would love to know which concepts are active in the deeper layers of the model while generating the solution.

Is there a concept of “epsilon” or “delta”?

What are their projections on each other?

> the best way to find a previous proof of a seemingly open problem on the internet is not to ask for it; it's to post a new proof

https://mehmetmars7.github.io/Erdosproblems-llm-hunter/probl...

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