The greatest common divisor of certain sets of binomial coefficients (Q1063021)

Let \(d(n;r,s)=\gcd \{\left( \begin{matrix} n\\ k\end{matrix} \right)| k=r,...,s\}\) where \(n\geq s\geq r\geq 0\). It is known due to B. Ram that for \(n\geq 2\), \(d(n;1,n-1)=p\) or 1 according as \(n=p^ a\) (p a prime) or not. In this paper, the authors solve completely the difficult problem of obtaining a formula for d(n;r,s). Theorem 1. For \(n\geq s\geq 1\), we have \[ d(n;1,s)=n/\{1^{\epsilon_ 1(n)},2^{\epsilon_ 2(n)},...,s^{\epsilon_ s(n)}\} \] where \(\{\) \(\}\) denotes the least common multiple and \(\epsilon_ j(m)=1\) or 0 according as \(j| m\) or not. - Theorem 2. If \(s\geq 2r\), then \(d(n;r+1,s)=\prod^{r}_{m=0}d(n- m;1,s)\). - Finally in case \(r<s<2r\), they prove that the right side of the identity in Theorem 2 must be multiplied by an additional factor and they also determine this factor.