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

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.