On binomial coefficients associated with Sierpiński and Riesel numbers

Abstract

In this paper, we investigate the existence of Sierpiński numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer $r$, there exist infinitely many Sierpiński numbers and Riesel numbers of the form $\left(\genfrac{}{}{0ex}{}{k}{r}\right)$. Let $S(x)$ be the number of positive integers $r$ satisfying $1\le r\le x$ for which $\left(\genfrac{}{}{0ex}{}{k}{r}\right)$ is a Sierpiński number for infinitely many $k$. We further show that the value $S(x)/x$ gets arbitrarily close to 1 as $x$ tends to infinity. Generalizations to base $a$-Sierpiński numbers and base $a$-Riesel numbers are also considered. In particular, we prove that there exist infinitely many positive integers $r$ such that $\left(\genfrac{}{}{0ex}{}{k}{r}\right)$ is simultaneously a base $a$-Sierpiński and base $a$-Riesel number for infinitely many $k$.