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

TL;DR

This survey comprehensively reviews Menon's identity, its proofs, generalizations, and analogs, highlighting various methods and historical context, serving as a detailed reference for researchers interested in this number theory topic.

Contribution

It provides detailed, self-contained proofs of Menon's identity, surveys its generalizations and analogs, and includes historical remarks and an updated references list.

Findings

Multiple proofs of Menon's identity using different methods

Survey of generalizations and analogs of Menon's identity

Historical overview and updated references included

Abstract

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