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

TL;DR

This paper generalizes Sierpiński's theorem using Lucas sequences, proving the existence of infinitely many parameters and integers for which certain sequences remain composite, extending the original result to broader algebraic structures.

Contribution

It introduces a generalization of Sierpiński's theorem involving Lucas sequences and establishes the existence of infinitely many such sequences and parameters with the same compositeness property.

Findings

Existence of infinitely many Lucas pairs with the property

Generalization to nonlinear versions with rational parameters

Extension of Sierpiński's theorem to broader sequences

Abstract

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 $n\ge 1$.