Generalized Sierpiński and Riesel numbers of the form $t{b}^t+\alpha$

Abstract

Let $b\ge 2$ be an integer. We call an integer $k$ a $b$-Sierpiński 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 $b$-Sierpiński and $b$-Riesel is called a $b$-Brier number. In this article, we show that for any integer $\alpha \ne 0$, there are infinitely many $b$-Sierpiński numbers and infinitely many $b$-Riesel numbers of the form $t{b}^t+\alpha$. We further show that when $b+1$ is not a power of $2$, there are infinitely $b$-Brier number of this form.