Generalizations of Menon's arithmetic identity

TL;DR

This paper presents elementary proofs of generalizations of Menon's arithmetic identity, expanding its applicability and providing deeper insights into its structure.

Contribution

It introduces new generalizations of Menon's identity and offers elementary proofs, enhancing understanding of this classical number theory result.

Findings

Extended Menon's identity to broader classes of residue sets

Elementary proofs simplify understanding of the generalizations

Potential applications in divisor and totient function analysis

Abstract

Menon's identity is $\sum_{a \in A}^m (a-1,m) = d(m) \varphi(m)$, where $A$ is a reduced set of residues modulo $m$. This paper contains elementary proofs of some generalizations of this result.