Proofs, generalizations and analogs of Menon's identity: a survey

Abstract

Menon's identity states that for every positive integer $n$ one has $\sum (a-1,n)=\phi (n)\tau (n)$, where $a$ runs through a reduced residue system (mod $n$), $(a-1,n)$ stands for the greatest common divisor of $a-1$ and $n$, $\phi (n)$ is Euler's totient function and $\tau (n)$ 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.

46 pages, revised, references updated

References in corpus (4)

• Menon-type identities again: A note on a paper by Li, Kim and Qiao
• A new Menon-type identity derived from group actions
• A generalization of Tóth identity in the ring of algebraic integers involving a Dirichlet Character
• Menon-type identities concerning subsets of the set $\{1,2,\dots ,n\}$

Cited by in corpus (1)

• Menon-type identities concerning subsets of the set $\{1,2,\dots ,n\}$