Comptes Rendus
no. 8
Number theory
The greatest common divisor of certain binomial coefficients

Presented by: the Editorial Board

Siao Hong1
1 Center for Combinatorics, Nankai University, Tianjin 300071, PR China
Comptes Rendus. Mathématique, Volume 354 (2016) no. 8, pp. 756-761.

Let m and n be positive integers. Let (mn)=m!n!(m−n)! denote the binomial coefficient indexed by m and n, where n! is the factorial of n. For any prime p, let νp(n) denote the largest nonnegative integer r such that pr divides n. In this paper, we use the p-adic method to show the following identity:

gcd⁡({(mnk):1≤k≤mn,gcd⁡(k,m)=1})=m∏primep|gcd⁡(m,n)pνp(n).

This extends greatly the identities obtained by Mendelsohn et al. in 1971 and by Albree in 1972, respectively.

Soient m et n deux entiers positifs. Soit (mn)=m!n!(m−n)! le coefficient binomial. Pour chaque nombre premier p, soit νp(n) le plus grand entier r tel que pr divise n. Dans cet article, nous montrons l'identité suivante :

gcd⁡({(mnk):1≤k≤mn,gcd⁡(k,m)=1})=m∏primep|gcd⁡(m,n)pνp(n).
Ceci améliore les identités obtenues par Mendelsohn et al. en 1971 et par Albree in 1972.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.06.001

Siao Hong 1

1 Center for Combinatorics, Nankai University, Tianjin 300071, PR China 
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Siao Hong. The greatest common divisor of certain binomial coefficients. Comptes Rendus. Mathématique, Volume 354 (2016) no. 8, pp. 756-761. doi : 10.1016/j.crma.2016.06.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.06.001/

[1] J. Albree The gcd of certain binomial coefficients, Math. Mag., Volume 45 (1972), pp. 259-261

[2] K. Ireland; M. Rosen A Classical Introduction to Modern Number Theory, Grad. Texts Math., vol. 84, Springer-Verlag, New York, 1990

[3] H. Joris; C. Oestreicher; J. Steinig The greatest common divisor of certain sets of binomial coefficients, J. Number Theory, Volume 21 (1985), pp. 101-119

[4] N.S. Mendelsohn; St. Olaf College Students, divisors of binomial coefficients, Amer. Math. Mon., Volume 78 (1971), pp. 201-202

[5] B. Ram Common factors of n!/m!(n−m)! (m=1,2,...,n−1), J. Indian Math. Club (Madras), Volume 1 (1909), pp. 39-43

[6] C. Soulé Secant varieties and successive minima, J. Algebraic Geom., Volume 13 (2004), pp. 323-341

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