The number of prime factors of binomial coefficients
Let \(\binom{n}{r}\) denote the binomial coefficients and let \(\omega\left(\binom{n}{r}\right)\) denote the number of distinct prime factors of \(\binom{n}{r}\). Then various interesting results involving \(\omega\left(\binom{n}{r}\right)\) are proved. It is best to quote the abstract. ``We investigate lower bounds for \[ \omega\left(\binom{n}{s}\right) - \omega\left(\binom{n}{r}\right) \] that are independent of \(n\). This difference is \(\ge -g(r)\) for all \(s\ge r\) and all \(n\ge 2s\) and is \(\ge 0\) for all \(s>r+ f(r)\) and all \(n\ge 2s\), where the functions \(g(r)\) and \(f(r)\) are \(o(r)\). These functions are evaluated exactly for \(r\le10\). In particular \(g(1) = f(1) =0\), so that \(\binom{n}{s}\) has at least as many distinct prime factors as \(n\) for all \(s\ge 1\) and \(n\ge s+1\). For the corresponding functions when finitely many exceptional values of \(n\) are allowed a method of Ramachandra gives the stronger estimate \(O(r^{\tfrac12 - \delta})\). Similar results are obtained with \(\Omega\) in place of \(\omega\).''
scientific article; zbMATH DE number 3769747