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Claim. The note A note on Problem #684, by Quanyu Tang and ChatGPT-5.2 Thinking, dated 19 January 2026 and linked from Tang's thread post of the same day, proves as its Theorem 1.2 that for every $\varepsilon>0$ there is $n_0(\varepsilon)$ such that for all $n\ge n_0(\varepsilon)$ the function $f(n)$ of Problem 684 satisfies

$$f(n)\le\bigl\lceil n^{12/17+\varepsilon}\bigr\rceil.$$

The proof writes $\binom nk=u(n,k)v(n,k)$, observes that for $k\ge\sqrt n$ every prime above $k$ divides $\binom nk$ at most once and only when a multiple of it lies in $(n-k,n]$, bounds $\log v(n,k)$ by sums of $\psi(n/t)-\psi((n-k)/t)$ over $t\le n/k$, and uses the short-interval asymptotic for $\psi$ at exponent $7/12$ from zero-density estimates. The note's Remark 1.3 says the proof was generated by ChatGPT-5.2 Thinking in an iterative process, with the prompts recorded in its appendix; the thread posts of 20 January 2026 add that Tang checked the argument, polished the presentation, and fed the model a passage of its own earlier reasoning to reach the exponent. The exponent $30/43$ that the site's remarks credit to Tang and ChatGPT is not in the note: the curator suggested in the thread on 19 January 2026 that the large-value estimates of Guth and Maynard improve the exponent to $30/43$, and to $2/3$ under the Riemann hypothesis or the density hypothesis, and Tang's post of 20 January 2026 says the note's method gives $f(n)\le n^{30/43+o(1)}$ with that input; no written proof of it was posted.

Submission note. Posted to the site's forum by Quanyu Tang on 19 January 2026:

Since many people are now using AI to explore problems on this site, I tried this problem using ChatGPT-5.2 Thinking and obtained a nontrivial upper bound:

$$f(n)\le \Bigl\lceil n^{\frac{12}{17}+\varepsilon}\Bigr\rceil \qquad > (\forall\,\varepsilon>0,\ n\ge n_0(\varepsilon)).$$

A full write-up is in this

note.

Verification. I audited the final lemma--proof write-up in multiple fresh and independent sessions of ChatGPT-5.2 Thinking. All audits agreed that the proof is logically correct, up to minor presentational clarifications. EDIT: I have also carefully reviewed the proof myself and found no issues with it.

How the result was obtained. The entire derivation was produced by the AI model, without human intervention in the mathematical reasoning or idea generation. I repeatedly asked ChatGPT-5.2 Thinking for stronger unconditional upper bounds for $f(n)$: it started from a basic bound $f(n)\le n/2$ and was iteratively strengthened, eventually yielding the polynomial improvement above. The full prompt history is recorded in the appendix of this note.

P.S. This is my first time trying to tackle a problem on this site using AI end-to-end, and I am not sure whether this kind of comment/approach is valuable to the community; I would be very happy to hear any feedback.

(The site has been updated to address this comment.)

Posted to the site's forum by Quanyu Tang on 20 January 2026:

Thanks for the comments! I agree with Bloom's point. Following the same method in my note, one can plug in the recent work of Guth and Maynard [GuMa24] to obtain the stronger unconditional bound $f(n) \le n^{30/43+o(1)}.$

Also, for clarity: the proof in the note was generated by the AI model; I carefully checked the argument myself, and only manually polished the introduction and some presentational details/technical steps in the proof. I did not share the raw chat logs because I used multiple independent chat sessions to audit each iteration of the argument, and most of those conversations were in Chinese (rather than English).

Reference. [GuMa24] Guth, Larry, and James Maynard. "New large value estimates for Dirichlet polynomials." arXiv preprint arXiv:2405.20552 (2024).

(The site has been updated to address this comment.)

Covers. The bound $f(n)\le\lceil n^{12/17+\varepsilon}\rceil$ for large $n$, and the bound $f(n)\le n^{30/43+o(1)}$ as stated in the thread. Both are superseded by the $(\log n)^2$ bound of Alexeev, Putterman, Sawhney, Sellke and Valiant 2026. The order of $f(n)$ is not determined.

Depends on. No page of this wiki; the argument rests on the cited zero-density estimate for primes in short intervals.

Standing. The note is an unrefereed research note on the author's repository, pinned above at the commit of its upload. The site's remarks credit Tang and ChatGPT with the $30/43$ bound and thank Tang, but the site labels the problem OPEN (page last edited 1 April 2026), so the remark is commentary on a partial result and not acceptance; the curator's thread comment of 20 January 2026 that the proof looks good to them, and Terence Tao's comment the same day classifying it as a partial result on the community's wiki of AI contributions, are commentary as well. The claim is claimed.