Problem 387

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claims/: The 2 claim pages of Problem 387, one per claimant's result; the problem's standing derives from them.

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Statement. Is there an absolute constant $c>0$ such that, for all $1\leq k< n$, the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn,n]$?

Status. The site labels the problem SOLVED, crediting Bui, Naprienko, Pratt and Zaharescu with the negative answer. The standing derived from the claim pages is solved, disproved: no constant $c>0$ works, by the accepted claim Bui, Naprienko, Pratt and Zaharescu 2026.

Source. erdosproblems.com/387, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #387, https://www.erdosproblems.com/387.

References.

• [BNPZ26] H. Bui, S. Naprienko, K. Pratt, and A. Zaharescu, Binomial coefficients with divisors avoiding an interval. arXiv:2605.21221 (2026).
• [Er78g] Erdős, Pál, On prime factors of binomial coefficients. II. Mat. Lapok (1978/82), 307-316.
• [Fa66] Faulkner, M., On a theorem of Sylvester and Schur. J. London Math. Soc. (1966), 107-110.
• [Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section B33 "Largest divisor of a binomial coefficient", printed p. 134: Erdős's conjecture that $\binom nk$ has a divisor between $cn$ and $n$ for any $c<1$ and $n$ sufficiently large. Library home: guy_2004_unsolved_problems_number_theory.
• [Sc58] Schinzel, A., Sur un problème de P. Erdős. Colloq. Math. (1958), 198-204.

Formalization. Statement in formal-conjectures, as erdos_387 : answer(False) ↔ … with sorry; no kernel-checked proof of the answer is recorded. Naprienko's repository holds Lean 4 proofs of a weaker form of the paper's covering proposition (with $B$ fixed, not the version the proof uses) and of Tao's residue-cover conjecture, resting on two analytic axioms; this corpus has not built them.

Current assessment

The dated site formulation above asks for one constant $c>0$ serving every $\binom nk$ with $1\leq k<n$. The answer is no: Bui, Naprienko, Pratt and Zaharescu 2026 exhibit infinitely many coefficients, with $k$ of order $(\log\log n)^{1/2}$, whose divisors all avoid an interval $(cn,n]$ with $c=c(n)\to0$. The same paper proves the opposite for large $k$: when $\exp((\log n)^{2/3+\varepsilon})\leq k\leq n/2$ and $n$ is large, $\binom nk$ has a divisor in $(n-n/(\log n)^{1/4},n]$, so the stronger form Guy records in section B33, a divisor in $(cn,n]$ for every $c<1$ and large $n$, holds in that range of $k$. The first version of the paper (2026-05-20) had the negative answer under the Generalized Riemann Hypothesis; the second (2026-06-30) is unconditional. The site's curator marked the problem solved on the unconditional version, and no journal publication is recorded. An unsigned note posted in the site's thread on 2026-05-29 by Przemek Chojecki, its argument credited to GPT-5.5 Pro, claims the same negative answer for every fixed window $(n/B,n]$ by pairing a fixed-$B$ covering construction with the first version's divisor propositions; Naprienko described it as a rewriting of his own construction, public since 2026-05-23, and it is a pending claim on its page.

Earlier results frame the question. Erdős had asked whether there is always a divisor in $(n-k,n]$, a stronger interval, and Schinzel answered negatively with $n=99215$, $k=15$ ([Sc58], library card Schinzel 1958); Schinzel's own conjecture, that every large $k$ which is not a prime power admits an $n$ with no divisor of $\binom nk$ in $(n-k,n]$, is problem B34 of [Gu04]. A divisor in $[n/k,n]$ always exists: $k\binom nk=n\binom{n-1}{k-1}$, so $n/\gcd(n,k)$ divides $\binom nk$, and $n/\gcd(n,k)\geq n/k$. Faulkner [Fa66] gives a prime divisor at least the least prime above $2k$ when $n$ is at least that prime, apart from $\binom92$ and $\binom{10}3$. Erdős wrote in [Er78g] that he expected a negative answer. Search scope, 2026-10-07: the site's problem page and discussion thread, the arXiv record of the paper with its three versions, and the formal-conjectures file. The paper's proofs are not checked here, and this page takes [Er78g], [Fa66] and [Sc58] from the site's and the library card's reports of them.

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.

• bui_2026_binomial_coefficients_divisors_avoiding_interval
• bui_2026_binomial_coefficients_divisors_avoiding_interval / corollary_1_6
• bui_2026_binomial_coefficients_divisors_avoiding_interval / question_1_1
• bui_2026_binomial_coefficients_divisors_avoiding_interval / theorem_1_2
• bui_2026_binomial_coefficients_divisors_avoiding_interval / theorem_1_4
• bui_2026_binomial_coefficients_divisors_avoiding_interval / theorem_5_1
• schinzel_1958_sur_un_probleme_de_p_erdos
• guy_2004_unsolved_problems_number_theory