Proofs, generalizations and analogs of Menon’s identity: a survey
Menon’s identity states that for every positive integer n one has ∑ (a − 1, n) = φ (n)τ(n), where a runs through a reduced residue system (mod n), (a − 1, n) stands for the greatest common divisor of a − 1 and n, φ(n) is Euler’s totient function, and τ(n) is the number of divisors of n. Menon’s iden...
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| Format: | Article |
| Language: | English |
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Sciendo
2023-11-01
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| Series: | Acta Universitatis Sapientiae: Mathematica |
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| Online Access: | https://doi.org/10.2478/ausm-2023-0009 |
| _version_ | 1797561921749123072 |
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| author | Tóth László |
| author_facet | Tóth László |
| author_sort | Tóth László |
| collection | DOAJ |
| description | Menon’s identity states that for every positive integer n one has ∑ (a − 1, n) = φ (n)τ(n), where a runs through a reduced residue system (mod n), (a − 1, n) stands for the greatest common divisor of a − 1 and n, φ(n) is Euler’s totient function, and τ(n) is the number of divisors of n. Menon’s identity has been the subject of many research papers, also in the last years. We present detailed, self contained proofs of this identity by using different methods, and point out those that we could not identify in the literature. We survey the generalizations and analogs, and overview the results and proofs given by Menon in his original paper. Some historical remarks and an updated list of references are included as well. |
| first_indexed | 2024-03-10T18:21:31Z |
| format | Article |
| id | doaj.art-027787e7f41048b6a915661f3d5c2c56 |
| institution | Directory Open Access Journal |
| issn | 2066-7752 |
| language | English |
| last_indexed | 2024-03-10T18:21:31Z |
| publishDate | 2023-11-01 |
| publisher | Sciendo |
| record_format | Article |
| series | Acta Universitatis Sapientiae: Mathematica |
| spelling | doaj.art-027787e7f41048b6a915661f3d5c2c562023-11-20T07:17:10ZengSciendoActa Universitatis Sapientiae: Mathematica2066-77522023-11-0115114219710.2478/ausm-2023-0009Proofs, generalizations and analogs of Menon’s identity: a surveyTóth László01Department of Mathematics, University of Pécs, Ifjúságútja 6, 7624 Pécs, HungaryMenon’s identity states that for every positive integer n one has ∑ (a − 1, n) = φ (n)τ(n), where a runs through a reduced residue system (mod n), (a − 1, n) stands for the greatest common divisor of a − 1 and n, φ(n) is Euler’s totient function, and τ(n) is the number of divisors of n. Menon’s identity has been the subject of many research papers, also in the last years. We present detailed, self contained proofs of this identity by using different methods, and point out those that we could not identify in the literature. We survey the generalizations and analogs, and overview the results and proofs given by Menon in his original paper. Some historical remarks and an updated list of references are included as well.https://doi.org/10.2478/ausm-2023-0009menon’s identityarithmetic functioneuler’s functionramanujan’s sumdirichlet characterdirichlet convolutionunitary convolutiongroup actionorbit counting lemman-even functionfinite fourier representation11a0711a2522f05 |
| spellingShingle | Tóth László Proofs, generalizations and analogs of Menon’s identity: a survey Acta Universitatis Sapientiae: Mathematica menon’s identity arithmetic function euler’s function ramanujan’s sum dirichlet character dirichlet convolution unitary convolution group action orbit counting lemma n-even function finite fourier representation 11a07 11a25 22f05 |
| title | Proofs, generalizations and analogs of Menon’s identity: a survey |
| title_full | Proofs, generalizations and analogs of Menon’s identity: a survey |
| title_fullStr | Proofs, generalizations and analogs of Menon’s identity: a survey |
| title_full_unstemmed | Proofs, generalizations and analogs of Menon’s identity: a survey |
| title_short | Proofs, generalizations and analogs of Menon’s identity: a survey |
| title_sort | proofs generalizations and analogs of menon s identity a survey |
| topic | menon’s identity arithmetic function euler’s function ramanujan’s sum dirichlet character dirichlet convolution unitary convolution group action orbit counting lemma n-even function finite fourier representation 11a07 11a25 22f05 |
| url | https://doi.org/10.2478/ausm-2023-0009 |
| work_keys_str_mv | AT tothlaszlo proofsgeneralizationsandanalogsofmenonsidentityasurvey |