Generalized Sierpiński and Riesel numbers of the form tbᵗ+α

Let b≥ 2 be an integer. We call an integer k a b-Sierpiński number if (k+1,b-1)=1 and k· bⁿ+1 is composite for all positive integers n. We similarly call k a b-Riesel number if (k-1,b-1)=1 and k· bⁿ-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 α≠ 0, there are infinitely many b-Sierpiński numbers and infinitely many b-Riesel numbers of the form tbᵗ+α. We further show that when $b+1$ is not a power of $2$, there are infinitely b-Brier number of this form.