On binomial coefficients associated with Sierpi\'{n}ski and Riesel numbers

TL;DR

This paper proves the existence of infinitely many binomial coefficients that are Sierpiński and Riesel numbers for all odd positive integers, and shows that the proportion of such numbers approaches 1 as the range increases.

Contribution

It establishes the infinite occurrence of binomial coefficients as Sierpiński and Riesel numbers for all odd integers and explores their generalizations to different bases.

Findings

Infinitely many binomial coefficients are Sierpiński and Riesel numbers for all odd r.

The ratio S(x)/x approaches 1 as x tends to infinity.

Existence of infinitely many r where binomial coefficients are both base a-Sierpiński and base a-Riesel numbers.

Abstract

In this paper, we investigate the existence of Sierpi\'{n}ski numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer $r$, there exist infinitely many Sierpi\'{n}ski numbers and Riesel numbers of the form $\binom{k}{r}$. Let $S(x)$ be the number of positive integers $r$ satisfying $1\leq r\leq x$ for which $\binom{k}{r}$ is a Sierpi\'{n}ski 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 $a$-Sierpi\'{n}ski numbers and base $a$-Riesel numbers are also considered. In particular, we prove that there exist infinitely many positive integers $r$ such that $\binom{k}{r}$ is simultaneously a base $a$-Sierpi\'{n}ski and base $a$-Riesel number for infinitely many $k$.