Mathematical Theory and Applications ›› 2022, Vol. 42 ›› Issue (1): 85-91.

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The Greatest Common Divisor of Certain Set of Binomial Coefficients

Xiao Jiaqi， Yuan Pingzhi*， Lin Xucan   

  1. School of Mathematical Science, South China Normal University, Guangzhou 510631, China
  • Online:2022-03-31 Published:2022-03-23
  • Contact: Pingzhi Yuan(1966-), Professor, PhD, E-mail: yuanpz@scnu.edu.cn
  • Supported by:
    This work is supported by National Natural Science Foundation of China (No. 12171163)

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Abstract:

In this paper, we prove that if n≥4 and a≥0 are integers satisfying a<n3, then

gcd({(nk):a<k<n−a})=∏n=pm+b(n,p), 0≤b(n,p)≤a,p,

where (nk)=n!k!(n−k)!, and the product in the right hand side runs through all primes p such that n=pm+b(n,p),m∈N and 0≤b(n,p)≤a.

As an application of our result, we give an answer to a problem in Hong [16].


Key words: Binomial coefficient , Greatest common divisor

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