Divisibility of the central binomial coefficient
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
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Abstract | References | Similar Articles | Additional Information
Abstract: We show that for every fixed , the set of
with
has a positive asymptotic density
and we give an asymptotic formula for
as
. We also show that
for some constant
. 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.
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Additional Information
Kevin Ford
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
Email:
ford@math.uiuc.edu
Sergei Konyagin
Affiliation:
Steklov Mathematical Institute, 8 Gubkin Street, Moscow, 119991, Russia
Email:
konyagin@mi-ras.ru
DOI:
https://doi.org/10.1090/tran/8183
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