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

Volume 8, Issue 1, February 2023     |     PP. 1-9      |     PDF (255 K)    |     Pub. Date: January 25, 2023
DOI: 10.54647/mathematics110379    105 Downloads     2636 Views  

Author(s)

Xing Zhu, School of Mathematical Science, Liaocheng University, No. 1, Hunan Road, Dongchangfu District, Liaocheng, 252059, Shandong, P.R. China
Chuanze Niu, School of Mathematical Science, Liaocheng University, No. 1, Hunan Road, Dongchangfu District, Liaocheng, 252059, Shandong, P.R. China

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 8, Issue 1, February 2023 | PP. 1-9. 10.54647/mathematics110379