Using Lucas Sequences to Generalize a Theorem of Sierpiński

Abstract

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