Using Lucas Sequences to Generalize a Theorem of Sierpi\'nski

In 1960, Sierpi\'nski proved that there exist infinitely many odd positive integers $k$ such that $k\cdot 2^n+1$ is composite for all positive integers $n$. In this paper, we prove some generalizations of Sierpi\'nski's theorem with $2^n$ replaced by expressions involving certain Lucas sequences $U_n(\alpha,\beta)$. In particular, we show the existence of infinitely many Lucas pairs $(\alpha,\beta)$, for which there exist infinitely many positive integers $k$, such that $k (U_n(\alpha,\beta)+(\alpha-\beta)^2)+1$ is composite for all integers $n\ge 1$. Sierpi\'nski's theorem is the special case of $\alpha=2$ and $\beta=1$. Finally, we establish a nonlinear version of this result by showing that there exist infinitely many rational integers $\alpha>1$, for which there exist infinitely many positive integers $k$, such that $k^2 (U_n(\alpha,1)+(\alpha-1)^2)+1$ is composite for all integers $n\ge 1$.