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

Ford, Kevin; Konyagin, Sergei

doi:10.1090/tran/8183

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
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Abstract: We show that for every fixed $\ell \in \mathbb{N}$ , the set of with $n^\ell \vert\binom {2n}{n}$ has a positive asymptotic density $c_\ell$ and we give an asymptotic formula for $c_\ell$ as $\ell \to \infty$ . We also show that $\char93 \{n\le x, (n,\binom {2n}{n})=1 \} \sim cx/\log x$ 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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