Malaysian Journal of Mathematical Sciences, May 2015, Vol. 9, No. 2
A Method for Determining $p$-Adic Orders of Factorials
Rafika Zulkapli, Kamel Ariffin Mohd Atan and Siti Hasana Sapar
Corresponding Email: rafikazulkapli@gmail.com
Received date: -
Accepted date: -
Abstract:
In this paper, with a prime $p$ the $p$-adic size of $n!$ where $n$ is a positive integer is determined for $ord_{p} n=0$ and $ord_{p} n > 0$ The discussion begins with the determination of the $p$-adic sizes of factorial functions $p^{\alpha}!$, $q^{\alpha}!$ and $(p^{\alpha}q^{\alpha})!$ with $\alpha, \beta > 0$ and $q$ a prime different from $p$. It is found that $ord_{p} p^{\alpha}! = \frac{p^{\alpha}-1}{p-1}$ with $\alpha > 0$. Results are then used to obtain the explicit form of $p$-adic sizes of $n$ from works of earlier authors. It is also found that the $p$-adic orders of $^{n}C_{r}=\frac{n!}{(n-r)!r!}$ is given by $ord_{p} {^{p^{\alpha}}}C_{p^{\theta}}=\frac{p^{\alpha}-p^{\theta}}{p-1}-\sum_{t=0}^{\left [ \frac{\ln\left ( p^{\alpha}-p^{\theta} \right )}{\ln p} \right ]} t \left ( \left [ p^{\alpha -t} - p^{\theta -t} \right ] - \left [ p^{\alpha -t-1} - p^{\theta -t-1} \right ]\right )$ where $n = p^{\alpha}$ and $r = p^{\theta}$ with $\alpha > \theta > 0$.
Keywords: Factorial functions, $p$-adic sizes
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