Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2023; 60(4): 863-872

Online first article July 19, 2023      Printed July 31, 2023

https://doi.org/10.4134/BKMS.b220166

Copyright © The Korean Mathematical Society.

On the greatest common divisor of binomial coefficients

Sunben Chiu, Pingzhi Yuan, Tao Zhou

South China Normal University; South China Normal University; South China Normal University

Abstract

Let $n\geqslant 2$ be an integer, we denote the smallest integer $b$ such that $\gcd\qty{\binom nk: b<k<n-b}>1$ as $b(n)$. For any prime $p$, we denote the highest exponent $\alpha$ such that $p^\alpha\mid n$ as $v_p(n)$. In this paper, we partially answer a question asked by Hong in 2016. For a composite number $n$ and a prime number $p$ with $p\mid n$, let $n=a_mp^m+r$, $0\leqslant r<p^m$, $0<a_m<p$. Then we have\\ \resizebox{\linewidth}{4.5mm}{ $\displaystyle v_p\qty(\gcd\qty{\binom nk: b(n)<k<n-b(n),\ (n,k)>1})= \begin{cases} 1,&a_m=1\text{ and }r=b(n), \\ 0,&\text{otherwise}. \end{cases} $}

Keywords: Binomial coefficient, greatest common divisor, $p$-adic valuation

MSC numbers: Primary 05A10, 11A05; Secondary 11D88, 11N05, 11A41

Supported by: Supported by National Science Foundation of China (No. 12171163).