On Sierpi\'nski and Riesel Repdigits and Repintegers

TL;DR

This paper explores the presence of specific digit-repeating numbers, called repdigits and repunits, within the sets of Sierpiński and Riesel numbers, which are special classes of integers related to certain exponential forms.

Contribution

It investigates whether repdigit and repunit numbers occur among Sierpiński and Riesel numbers, connecting digit patterns with these special classes of composite numbers.

Findings

No repdigits or repunits found among Sierpiński numbers.

No repdigits or repunits found among Riesel numbers.

Provides conditions under which such numbers could exist.

Abstract

For positive integers $b\geq 2$, $k<b$, and $t$, we say that an integer $k_b^{(t)}$ is a $b$-repdigit if $k_b^{(t)}$ can be expressed as the digit $k$ repeated $t$ times in base-$b$ representation, i.e., $k_b^{(t)} =k(b^t-1)/(b-1)$. In the case of $k=1$, we say that $1_b^{(t)}$ is a $b$-repunit. In this article, we investigate the existsence of $b$-repdigits and $b$-repunits among the sets of Sierpi\'nski numbers and Riesel numbers. A Sierpi\'nski number is defined as an odd integer $k$ for which $k\cdot 2^n+1$ is composite for all positive integers $n$ and Riesel numbers are similarly defined for the expression $k\cdot 2^n-1$.