Problem 684

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claims/: The 3 claim pages of Problem 684, one per claimant's result; the problem's standing derives from them.

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Statement. For $0\leq k\leq n$ write

$$\binom{n}{k} = uv$$

where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$.

Let $f(n)$ be the smallest $k$ such that $u>n^2$. Give bounds for $f(n)$.

Status. Open. The site labels the problem OPEN (page last edited 1 April 2026), its remarks recording the bounds of [APSSV26] and of Tang and ChatGPT and crediting no solution. The standing derives from the claim pages, three pending partial claims, none of which determines the order of $f(n)$: Tang 2026, a research note with the bound $f(n)\le\lceil n^{12/17+\varepsilon}\rceil$ and the thread's exponent $30/43$; Alexeev, Putterman, Sawhney, Sellke and Valiant 2026, an arXiv preprint with $f(n)\le(24/(\pi^2-6)+o(1))(\log n)^2$ for all large $n$ and $f(n)\ge(1/2+o(1))\log n$ along a sequence; and Bae 2026, an arXiv preprint with a Lean 4 proof that $f(n)\ge(1/2-o(1))\log n\,\log\log n/\log\log\log n$ for infinitely many $n$, which the site's curator relabeled partial because the order of $f(n)$ stays undetermined. So the problem is open with no settling or pending full claim. Known Results below record the density-one results, which have no claim pages.

Source. erdosproblems.com/684, accessed 2026-09-04 and 2026-10-07 (problem page last edited 1 April 2026). Cite as: T. F. Bloom, Erdős Problem #684, https://www.erdosproblems.com/684.

References.

• [APSSV26] B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant, Short proofs in combinatorics and number theory. arXiv:2603.29961 (2026).

Formalization. No formal-conjectures statement file exists for the problem. Bae's Lean 4 development, not built or audited here, is linked at a pinned tag from his claim page.

Current assessment

No assessment of the mathematics is recorded. The Status sentence gives the site's label and the standing derived from the claim pages. The notes under Known Results record outside results and claims and are not independently reviewed; this page records no assessment of proof coverage. Two results bound $f(n)$ only outside a set of density zero, Sothanaphan's notes of 2 April 2026 and Li's preprint of 6 June 2026 below; neither settles an instance of the worst-case question, which asks for bounds on $f(n)$ at every $n$, so neither has a claim page and both are recorded here.

Known Results

• Upper bound (recorded in the site's remarks, page last edited 1 April 2026; [APSSV26], v1 31 March 2026, v2 2 April 2026, unrefereed, its argument attributed by the paper and by the site to an internal OpenAI model, with the authors editing the write-up): $f(n)\le(24/(\pi^2-6)+o(1))(\log n)^2$ for all large $n$, and $f(n)\ge(1/2+o(1))\log n$ along a sequence of $n$; recorded on the claim page Alexeev, Putterman, Sawhney, Sellke and Valiant 2026. Library home: alexeev_2026_short_proofs_combinatorics_number_theory.
• Lower bound for infinitely many $n$, machine-checked, unrefereed: Ji Ho Bae, arXiv:2604.23784 (v3, 3 September 2026; the arguments of v1 and v2 were withdrawn by the author as unjustified), recorded on the claim page Bae 2026: $f(n)>(1/2-\varepsilon)\log n\,\log\log n/\log\log\log n$ for infinitely many $n$, so $f(n)/\log n$ is unbounded. A Lean 4 proof is registered on the Palomar registry as PALOMAR-2026-09-03-000006 (3 September 2026, trust level high, standard axioms reported; challenge statement in jidodat/erdos684-lean). The registry replays the proof in the Lean kernel and compares it with a challenge statement; its own description says that it certifies neither novelty nor the match between the formal and informal statements and is not peer review, and no refereed version, site acceptance or independent review was found. The author's proof claim on the site's proof-claims tab (submitted 2026-09-04) says that the two bounds together answer the problem's request for bounds on $f(n)$ and that the exact extremal order is open; the site's maintainer relabeled it a partial claim the same day, since how fast $f(n)$ grows is still uncertain, and asked whether $f(n)\gg(\log n)^2$ infinitely often. This corpus has not built the development.
• Almost all $n$ (unrefereed): Li, arXiv:2606.08216 (v1, 6 June 2026), $f(n)=(2/(1-\gamma)+o(1))\log n$ outside a set of density zero, which its abstract calls not a pointwise resolution of the worst-case problem; no claim page, for the reason given in the Current assessment. Library home: li_2026_erdos_problem_684_at_density_one.
• Almost all $n$, earlier (unrefereed): $f(n)\le(4/(1-\gamma)+o(1))\log n=(9.461\ldots+o(1))\log n$ for almost all $n$, in Nat Sothanaphan's notes Averaged logarithmic bounds for the binomial threshold problem of 2 April 2026, linked from his thread post of the same day; the notes' disclaimer says GPT-5.4 Thinking generated them in a near-autonomous process. The notes prove the bound for every threshold $n^\beta$, with constant $2\beta/(1-\gamma)$, by averaging over $n$ as a thread comment of 1 April 2026 had suggested; no claim page, for the reason given in the Current assessment.
• Superseded polynomial bounds: $f(n)\le\lceil n^{12/17+\varepsilon}\rceil$ for large $n$, proved in the note of Quanyu Tang and ChatGPT-5.2 Thinking of 19 January 2026, and $f(n)\le n^{30/43+o(1)}$ through Guth and Maynard's large-value estimates, stated in the thread and credited by the site's remarks to Tang and ChatGPT; recorded on the claim page Tang 2026.

Search scope. As of 2026-10-05 the site showed OPEN (last edited 1 April 2026) and did not mention Bae's or Li's results; the community database says open and unformalized; there is no conjectures.io or formal-conjectures entry. The open gap is the worst-case order of $f(n)$, between $\log n\,\log\log n/\log\log\log n$ and $(\log n)^2$. The frontmatter derives from the claim pages: the question asks for bounds, no source determines the order, and the three partial claims are pending, since the site's remarks on an open problem and a registry record are not acceptance under the anatomy's rule.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• li_2026_erdos_problem_684_at_density_one
• li_2026_erdos_problem_684_at_density_one / corollary_1_2
• li_2026_erdos_problem_684_at_density_one / proposition_5_3
• li_2026_erdos_problem_684_at_density_one / theorem_1_1
• li_2026_erdos_problem_684_at_density_one / theorem_1_3
• alexeev_2026_short_proofs_combinatorics_number_theory
• alexeev_2026_short_proofs_combinatorics_number_theory / theorem_2_1