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A NEW MENON’S IDENTITY FROM GROUP ACTIONS

Part of: Elementary number theory Sequences and sets

Published online by Cambridge University Press:  28 November 2018

YU-JIE WANG Open the ORCID record for YU-JIE WANG [Opens in a new window]  and
CHUN-GANG JI Open the ORCID record for CHUN-GANG JI [Opens in a new window]
YU-JIE WANG
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, PR China email [email protected]
CHUN-GANG JI*
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, PR China email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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Let $n$ be a positive integer. We obtain new Menon’s identities by using the actions of some subgroups of $(\mathbb{Z}/n\mathbb{Z})^{\times }$ on the set $\mathbb{Z}/n\mathbb{Z}$. In particular, let $p$ be an odd prime and let $\unicode[STIX]{x1D6FC}$ be a positive integer. If $H_{k}$ is a subgroup of $(\mathbb{Z}/p^{\unicode[STIX]{x1D6FC}}\mathbb{Z})^{\times }$ with index $k=p^{\unicode[STIX]{x1D6FD}}u$ such that $0\leqslant \unicode[STIX]{x1D6FD}<\unicode[STIX]{x1D6FC}$ and $u\mid p-1$, then

$$\begin{eqnarray}\mathop{\sum }_{x\in H_{k}}(x-1,p^{\unicode[STIX]{x1D6FC}})=\frac{\unicode[STIX]{x1D711}(p^{\unicode[STIX]{x1D6FC}})}{k}\bigg(1+k(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FD})+u\frac{p^{\unicode[STIX]{x1D6FD}}-1}{p-1}\bigg),\end{eqnarray}$$
where $\unicode[STIX]{x1D711}(n)$ is the Euler totient function.

Keywords

arithmetical sumsMenon’s identitygroup action

MSC classification

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 11B13: Additive bases, including sumsets
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 3 , June 2019 , pp. 369 - 375
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

This work was partially supported by the Grant No. 11471162 from NNSF of China and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20133207110012).

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