Home > Journals > SCIREA Journal of Mathematics > Archive > Paper Information

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    7 Downloads     121 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. Vol. 8 , No. 1 , 2023 , pp. 1 - 9 . https://doi.org/10.54647/mathematics110379

Submit A Manuscript
Review Manuscripts
Join As An Editorial Member
Most Views
Article
by Sergey M. Afonin
3057 Downloads 61980 Views
Article
by Jian-Qiang Wang, Chen-Xi Wang, Jian-Guo Wang, HRSCNP Research Team
15 Downloads 51279 Views
Article
by Syed Adil Hussain, Taha Hasan Associate Professor
2418 Downloads 25002 Views
Article
by Omprakash Sikhwal, Yashwant Vyas
2486 Downloads 21039 Views
Article
by Munmun Nath, Bijan Nath, Santanu Roy
2364 Downloads 20643 Views
Upcoming Conferences