English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

In 1960, Sierpi\'nski proved that there exist infinitely many odd positiveintegers $k$ 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 theoremwith $2^n$ replaced by expressions involving certain Lucas sequences$U_n(\alpha,\beta)$. In particular, we show the existence of infinitely manyLucas pairs $(\alpha,\beta)$, for which there exist infinitely many positiveintegers $k$, such that $k (U_n(\alpha,\beta)+(\alpha-\beta)^2)+1$ is compositefor 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 thisresult by showing that there exist infinitely many rational integers$\alpha>1$, for which there exist infinitely many positive integers $k$, suchthat $k^2 (U_n(\alpha,1)+(\alpha-1)^2)+1$ is composite for all integers $n\ge1$.

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