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arXiv · 1909.03903

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

Kevin Ford·Sergei Konyagin
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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 $\# \{n\le x, (n,\binom{2n}{n})=1 \} \sim cx/\log x$ for some constant $c$. One novelty is a method to capture the effect of large prime factors of integers in general sequences.

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Kevin Ford, Sergei Konyagin. 2020-05-02. Divisibility of the central binomial coefficient $\binom{2n}{n}$. https://arxiv.org/abs/1909.03903

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