Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Divisibility of the central binomial coefficient $\binom {2n}{n}$


Authors: Kevin Ford and Sergei Konyagin
Journal: Trans. Amer. Math. Soc. 374 (2021), 923-953
MSC (2020): Primary 05A10, 11B65; Secondary 11N25
DOI: https://doi.org/10.1090/tran/8183
Published electronically: December 3, 2020
MathSciNet review: 4196383
Full-text PDF

Abstract | Similar Articles | Additional Information

Abstract: We show that for every fixed $\ell \in \mathbb {N}$, the set of $n$ with $n^\ell |\binom {2n}{n}$ has a positive asymptotic density $c_\ell$ and we give an asymptotic formula for $c_\ell$ as $\ell \to \infty$. We also show that $\# \{n\leqslant x, (n,\binom {2n}{n})=1 \} \sim cx/\log x$ for some constant $c$. We use results about the anatomy of integers and tools from Fourier analysis. One novelty is a method to capture the effect of large prime factors of integers in general sequences.


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2020): 05A10, 11B65, 11N25

Retrieve articles in all journals with MSC (2020): 05A10, 11B65, 11N25


Additional Information

Kevin Ford
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
MR Author ID: 325647
ORCID: 0000-0001-9650-725X
Email: ford@math.uiuc.edu

Sergei Konyagin
Affiliation: Steklov Mathematical Institute, 8 Gubkin Street, Moscow, 119991, Russia
MR Author ID: 188475
Email: konyagin@mi-ras.ru

Received by editor(s): September 9, 2019
Received by editor(s) in revised form: February 2, 2020
Published electronically: December 3, 2020
Additional Notes: The first author was supported in part by National Science Foundation Grant DMS-1802139
The authors thank the anonymous referee for many helpful comments.
Article copyright: © Copyright 2020 American Mathematical Society