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 $n\ge 1$.