A generalization of Menonʼs identity

Abstract

In this note we give a generalization of the well-known Menon's identity. This is based on applying the Burnside's lemma to a certain group action.

Ideal Chains in Residually Finite Dedekind Domains

“…where σ r−1 (n) = d|n d r−1 . Tǎrnǎuceanu [9] discussed an open problem from [8,Section 2] and Li and Kim [2] extended Tǎrnǎuceanu's results. Let D be a Dedekind domain such that the residue class ring D/n is finite for each nonzero ideal n. Then D is called a residually finite Dedekind domain.…”

“…As an application, we obtain some new representations of (1.1) and (1.2) (Remarks 4.4 and 4.5). In Sections 5 and 6, we obtain generalisations in residually finite Dedekind domains of the Menon-type identities in [2,9] (Theorems 5.2 and 6.2).…”

“….. , d r are all invariant factors of the matrixA − E r in D p satisfying d 1 | d 2 | • • • | d r .Remark 5.6. If D = Z, then Corollary 5.5 reduces to the main theorem of[9].…”

Counting invertible sums of squares modulo $n$ and a new generalization of Euler's totient function

“…1≤j≤n gcd(j,n)=1 gcd(j − 1, n) = ϕ(n)d(n), valid for every n ∈ N, where d(n) denotes the number of divisors of n. This identity has been generalized in several ways. See, for example [11,12,17,19]. Also, 1≤j≤n gcd(j,n)=1 gcd(j 2 − 1, n) = ϕ(n)h(n),…”

The Number of Cyclic Subgroups of Finite Abelian Groups and Menon’s Identity

“…The generalisations involve additive and multiplicative characters (see [7,22,23]), arithmetical functions of several variables (see [20]), actions of subgroups of GL r (Z n ) (see [5,6,19]) and residually finite Dedekind domains (see [10,11]).…”