The $p$-adic valuation of $C_n(l)=\prod_{k=1}^{n}\prod_{j=0}^{l}(2k+2j-1)$

Author(s)

Abstract
The formula of the $p$-adic valuation of the sequence $C_n(l)=\prod_{k=1}^{n}\prod_{j=0}^{l}(2k+2j-1)$ is studied. It is proved that $C_n(l)$ is not a square if $l$ is even and $n\ge \max\{2l,\frac{3l+19}{5}\}$. It is proved also that there are many squares in the sequences $C_n(1)$ and $C_n(3)$, while there are no squares in the sequences $C_n(5)$, $C_n(7)$ and $C_n(9)$.

Keywords
p-adic valuation, Legendre’s formula, Pell equation

Cite this paper
Xing Zhu, Chuanze Niu, The $p$-adic valuation of $C_n(l)=\prod_{k=1}^{n}\prod_{j=0}^{l}(2k+2j-1)$ , SCIREA Journal of Mathematics. Volume 10, Issue 3, June 2025 | PP. 1-9. 10.54647/mathematics110379