• Exact divisibility by powers of the Pell and Associated Pell numbers

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/131/0020

• # Keywords

Exact divisibility; Pell number; Associated Pell number; $p$-adic valuation; order of appearance.

• # Abstract

Let $P_n$ be the $n$-th Pell number and $Q_n$ be the $n$-th Associated Pell number. We obtain certain divisibility and exact divisibility results for the powers of the Pell and Associated Pell numbers by applying the concept of $p$-adic valuation of the numbers. In particular, we show that $P^k_n\mid\mid m$ if and only if $P^{k+1}_n\mid\mid P_{nm}$ for all $m, n \geq 2$ and $k \geq 1$. In addition, $Q^k_n\mid\mid m$ if and only if $Q^{k+1}_n\mid\mid Q_{nm}$ for all $m, n \geq 2$ and $k \geq 1$ is also proved.

• # Author Affiliations

1. Department of Mathematics, National Institute of Technology, Rourkela 769 001, India

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019