On Number of Ordered Pair of Positive Integers with Given Least Common Multiple
Abstract views: 119
/ 
PDF downloads: 56
Abstract
In this article, we present an expression for the number of ordered pairs of positive integer $m$ with prime factorization $p_1^{\alpha_1} \cdot p_2^{\alpha_3} \ldots p_n^{\alpha_n}$, and en route introduce an identity viz:
%\begin{equation*}
\begin{align*}
&\prod_{i=1}^n (2 \alpha_i+1) = \sum (\alpha_1+1) \alpha_2 \ldots \alpha_n + \sum (\alpha_1+1)(\alpha_2+1) \alpha_3 \ldots \alpha_n + \\
& \cdots + \sum (\alpha_1+1)(\alpha_2+1) \ldots (\alpha_r+1) \alpha_{r+1} \ldots \alpha_n + \cdots + \alpha_1 \alpha_2 \ldots \alpha_n
\end{align*}
%\end{equation*}
%\medskip
Moreover, we investigate the asymptotic behaviour of the mean and variance of the relative number of order pairs with some given least common multiple $m$. The notion of ordered pair is widely used in the fields of geometry, statistics, computing and programming languages.
References
A. Tucker, Applied Combinatorics, Wiley, USA, 2012.
B. Sury, “Lower Bound for the Least Common Multiple,” The American Mathematical Monthly, 126:10 (2019), pp. 940–942, https://doi.org/10.1080/00029890.2019.1651179.
C. Sanna, “On the least common multiple of random q-integers,” Research in Number Theory, 7(1) (2021), Article 16, https://doi.org/10.1007/s40993-021-00242-4.
C. T. Long, Elementary Introduction to Number Theory, Prentice Hall, Englewood Cliffs, NJ, 1987.
D. M. Burton, Elementary Number Theory, 6th ed., Tata McGraw-Hill, 2007.
G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, Cambridge, 1995.
J. Buhler, Cryptography, Cambridge University Press, 2020.
J. O’Rourke, Computational Geometry in C, 2nd ed., Cambridge University Press, 1998.
K. Ford and T. Tao, “The distribution of values of ϕr (n),” American Journal of Mathematics 136 (2014), no. 3, pp. 645–669.
M. Z. Spivey, The Art of Proving Binomial Identities, CRC Press, p. 71, ISBN 1351215809, 2019.
R. C. Vaughan, “On the distribution of αp(n),” Mathematika 51 (2004), no. 1-2, pp. 135–184.
R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis, Wiley, New York, 2000.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Results in Nonlinear Analysis
This work is licensed under a Creative Commons Attribution 4.0 International License.
