On Sierpiński and Riesel Repdigits and Repintegers

Abstract

For positive integers b≥2 , k<b, and t we say that an integer k₆⁽ᵗ⁾ is a b-repdigit if k₆⁽ᵗ⁾ can be expressed as the digit k repeated t times in base-b representation, i.e., k₆⁽ᵗ⁾ =k(bᵗ-1)/(b-1). In the case of k=1, we say that 1₆⁽ᵗ⁾ is a b-repunit. In this article, we investigate the existsence of b-repdigits and b-repunits among the sets of Sierpiński numbers and Riesel numbers. A Sierpiński number is defined as an odd integer k for which k⋅2ⁿ+1 is composite for all positive integers $n$ and Riesel numbers are similarly defined for the expression k⋅2ⁿ-1.