Generalized Sierpi\'{n}ski and Riesel numbers of the form $tb^t+\alpha$

TL;DR

This paper proves the existence of infinitely many generalized Sierpiński, Riesel, and Brier numbers of the form $tb^t + ext{constant}$ for various bases, expanding the understanding of these special composite number classes.

Contribution

It establishes the infinite existence of such numbers for any nonzero constant, generalizing classical cases and including new conditions for Brier numbers.

Findings

Infinitely many $b$-Sierpiński numbers of the form $tb^t + ext{constant}$ exist.

Infinitely many $b$-Riesel numbers of the form $tb^t + ext{constant}$ exist.

When $b+1$ is not a power of 2, infinitely many $b$-Brier numbers of this form exist.

Abstract

Let $b\geq 2$ be an integer. We call an integer $k$ a $b$-Sierpi\'{n}ski number if $\gcd(k+1,b-1)=1$ and $k\cdot b^n+1$ is composite for all positive integers $n$. We similarly call $k$ a $b$-Riesel number if $\gcd(k-1,b-1)=1$ and $k\cdot b^n-1$ is composite for all positive integers $n$. An integer that is simultaneously $b$-Sierpi\'{n}ski and $b$-Riesel is called a $b$-Brier number. In this article, we show that for any integer $\alpha\neq 0$, there are infinitely many $b$-Sierpi\'{n}ski numbers and infinitely many $b$-Riesel numbers of the form $tb^t+\alpha$. We further show that when $b+1$ is not a power of $2$, there are infinitely $b$-Brier number of this form.