A generalization of Menon’s identity to higher exponent

Yan Li; Daeyeoul Kim; Rui Qiao

doi:10.5486/PMD.2019.8433

A generalization of Menon’s identity to higher exponent

Yan Li
, Daeyeoul Kim
, Rui Qiao

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

2 Scopus citations

Abstract

In this note, we shall explicitly compute the following sum X gcd(a^` − 1, b1, . . ., b_k, n), 1≤a,b1,...,bk≤n gcd(a,n)=1 where n ≥ 1, k ≥ 0, l ≥ 1 are integers. Our results extend Menon’s identity and Sury’s identity (i.e., ` = 1 in the above summation) to higher exponents. Note that in the case k = 0, some of our results are recovered by the results of [21].

Original language	English
Pages (from-to)	467-475
Number of pages	9
Journal	Publicationes Mathematicae Debrecen
Volume	94
Issue number	3-4
DOIs	https://doi.org/10.5486/PMD.2019.8433
State	Published - 2019

Keywords

Chinese remainder theorem
Dirichlet character
Dirichlet convolution
Divisor function
Euler’s totient function
Menon’s identity

Quacquarelli Symonds(QS) Subject Topics

Mathematics

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10.5486/PMD.2019.8433

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@article{1b1dad9308c34fdf8a1dd8dbddfad1eb,

title = "A generalization of Menon{\textquoteright}s identity to higher exponent",

abstract = "In this note, we shall explicitly compute the following sum X gcd(a` − 1, b1, . . ., bk, n), 1≤a,b1,...,bk≤n gcd(a,n)=1 where n ≥ 1, k ≥ 0, l ≥ 1 are integers. Our results extend Menon{\textquoteright}s identity and Sury{\textquoteright}s identity (i.e., ` = 1 in the above summation) to higher exponents. Note that in the case k = 0, some of our results are recovered by the results of [21].",

keywords = "Chinese remainder theorem, Dirichlet character, Dirichlet convolution, Divisor function, Euler{\textquoteright}s totient function, Menon{\textquoteright}s identity",

author = "Yan Li and Daeyeoul Kim and Rui Qiao",

year = "2019",

doi = "10.5486/PMD.2019.8433",

language = "English",

volume = "94",

pages = "467--475",

journal = "Publicationes Mathematicae Debrecen",

issn = "0033-3883",

number = "3-4",

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TY - JOUR

T1 - A generalization of Menon’s identity to higher exponent

AU - Li, Yan

AU - Kim, Daeyeoul

AU - Qiao, Rui

PY - 2019

Y1 - 2019

N2 - In this note, we shall explicitly compute the following sum X gcd(a` − 1, b1, . . ., bk, n), 1≤a,b1,...,bk≤n gcd(a,n)=1 where n ≥ 1, k ≥ 0, l ≥ 1 are integers. Our results extend Menon’s identity and Sury’s identity (i.e., ` = 1 in the above summation) to higher exponents. Note that in the case k = 0, some of our results are recovered by the results of [21].

AB - In this note, we shall explicitly compute the following sum X gcd(a` − 1, b1, . . ., bk, n), 1≤a,b1,...,bk≤n gcd(a,n)=1 where n ≥ 1, k ≥ 0, l ≥ 1 are integers. Our results extend Menon’s identity and Sury’s identity (i.e., ` = 1 in the above summation) to higher exponents. Note that in the case k = 0, some of our results are recovered by the results of [21].

KW - Chinese remainder theorem

KW - Dirichlet character

KW - Dirichlet convolution

KW - Divisor function

KW - Euler’s totient function

KW - Menon’s identity

UR - https://www.scopus.com/pages/publications/85071001497

U2 - 10.5486/PMD.2019.8433

DO - 10.5486/PMD.2019.8433

M3 - Journal article

AN - SCOPUS:85071001497

SN - 0033-3883

VL - 94

SP - 467

EP - 475

JO - Publicationes Mathematicae Debrecen

JF - Publicationes Mathematicae Debrecen

IS - 3-4

ER -

A generalization of Menon’s identity to higher exponent

Abstract

Keywords

Quacquarelli Symonds(QS) Subject Topics

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