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

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      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

       

      G K PANDA1 ASIM PATRA1

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

       
  • Proceedings – Mathematical Sciences | News

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