Open Access

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

   | Nov 15, 2023


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.

Publication timeframe:
2 times per year
Journal Subjects:
Mathematics, General Mathematics