Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 731

claims/: The 1 claim page of Problem 731, one per claimant's result; the problem's standing derives from them.

Statement. Find some reasonable function $f(n)$ such that, for almost all integers $n$ , the least integer $m$ such that $m\nmid \binom{2n}{n}$ satisfies

m\sim f(n).

Formulation. The site does not define "reasonable", and without some such restriction the request would be met trivially by taking $f(n)$ to be the least non-divisor itself. The page reads the question as its source does. [EGRS75] (p. 91) state without proof that, for every fixed $\epsilon>0$ and outside a set of density $0$ , the least non-divisor $A(n)$ of $\binom{2n}{n}$ satisfies $\exp((\log n)^{1/2-\epsilon})<A(n)<\exp((\log n)^{1/2+\epsilon})$ . They add that improving this would be easy but that an asymptotic formula looks hard. The question asks for such a formula: an explicit $f$ with $A(n)/f(n)\to1$ outside a set of density $0$ . The pending claim reads "reasonable" as dyadic regularity, a class that contains the usual explicit formulas, and asserts that no such $f$ exists.

Status. OPEN: the site's label, on a page last edited 19 October 2025, before the one claim. Eric Li's full claim, posted as an arXiv preprint on 2026-06-27 and submitted to the site's proof-claims tab on 2026-07-17, a resolution under Li's reading of "reasonable" as dyadic regularity, with a Lean development produced by Aristotle (Harmonic), is pending on the Li claim page. The derived standing, claimed and disproved, departs from the label because that full claim is pending: it asserts that no dyadically regular $f$ has the least non-divisor asymptotic to $f(n)$ for almost all $n$ , a negative answer under that reading.

Source. erdosproblems.com/731, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #731, https://www.erdosproblems.com/731.

References.

[EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\sp{2n}\sb{n})$ . Math. Comp. (1975), 83-92.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

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.

Linked from (14)

Li's resolution under dyadic regularity, with a Lean development problems/factorials_binomials/E0731/claims Factorials and Binomial Coefficients factorials_binomials/erdos_1975_prime_factors Inequality (8): the least integer not dividing C(2n,n) lies between exp((log n)^{1/2 -+ epsilon}) for almost all n Theorem 4: the number of integers up to n^alpha that do not divide C(2n,n)Theorem 5: few n fail to have C(2n,n) divisible by an m with only small prime-power factors factorials_binomials/li_2026_resolution_erdos_problem_731_under_dyadic Corollary 1.7 (p. 6): the global density-tight scale of the least non-divisor of the central binomial coefficient Lemma 2.1 (p. 7): the least non-divisor of the central binomial coefficient is a prime power Theorem 1.10 (p. 7): no dyadically regular asymptotic equivalent of the least non-divisor of the central binomial coefficient Theorem 1.3 (p. 5): dyadic mesoscopic tail bounds for the least non-divisor of the central binomial coefficient Theorem 1.4 (p. 5): dyadic nonconcentration of the least non-divisor of the central binomial coefficient Theorem 5.2 (pp. 13--14): moving-base strip variance for smoothed carry-free prime counts

Graph