一些二项式系数的最大公因数

The Greatest Common Divisor of Certain Set of Binomial Coefficients

本文证明:若n≥4和a≥0为整数且满足a<n/3,则gcd({(nk):a<k<n-a})=∏_(n=p^(m)+b(n,p),0≤a,)p,其中(nk)=n1/k1(n-k),右边的连乘积遍历所有满足n=p^(m)+b(n,p),m∈N和0≤b(n,p)≤a的素数p.作为上述结论的一个应用,我们回答洪[3]文中的一个问题.

In this paper,we prove that if n≥4 and a≥0 are integers satisfying a<n/3,then gcd({(nk):a<k<n-a})=∏_(n=p^(m)+b(n,p),0≤a,)p where(nk)n1/k1(n-k).and the product in the right hand side runs through all primes p such that n=p^(m)+b(n,p),m∈N abd 0≤b(n,p)≤a.As an application of our result,we give an answer to a problem in Hong[3].