As usual, Z,Q,C denote the ring of integers, the field of rational numbers and the field of complex numbers respectively. Let us fix a primitive cube root of unity ζ3 = −1+ √ −3 2 ∈ C. Let Q(ζ3) = Q( √ −3) be the third cyclotomic field and Z[ζ3] = Z +Z · ζ3 its ring of integers. We write λ for the (principal) maximal ideal (1− ζ3) · Z[ζ3] of Z[ζ3]. It is known [11, Th. 5 on p. 176] (see also [5...

For any positive integers n and r, let $$p_r(n)$$ denotes the number of partitions where each part has r distinct colours. Many authors studied partition function for particular values r. In this paper, we prove some general congruences modulo 5 7 colour by considering To employ q-series identities which are also in spirit Ramanujan.

For g, n coprime integers, let `g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of `g(n) as n ≤ x ranges over integers coprime to g, and x tending to infinity. Assuming the Generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptot...

For g, n coprime integers, let `g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of `g(n) as n ≤ x ranges over integers coprime to g, and x tending to infinity. Assuming the Generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptot...

We introduce a gcd-sum function involving regular integers (mod n) and prove results giving its minimal order, maximal order and average order.

The commuting graph of a group G, denoted by Γ(G), is the simple undirected graph whose vertices are the non-central elements of G and two distinct vertices are adjacent if and only if they commute. Let Zm be the commutative ring of equivalence classes of integers modulo m. In this paper we investigate the connectivity and diameters of the commuting graphs of GL(n,Zm) to contribute to the conje...

We shall consider a distribution property of sequences of integers. Let us denote a (a„)nGM an infinite sequence of integers. For integers N > 1, m > 2, and j (0 < J < 77Z — 1), let us define AN(j9 m9 a) as the number of terms among al3 a2S ...9aN satisfying the congruence an = j (mod m) . A sequence a (an)nEH±s said to be uniformly distributed modulo m (u.d. mod m) if, for every J = 0, 1, ...,...

a polynomial   1 2 ( , , , ) n f x x x is called multilinear if it is homogeneous and linear in every one of its variables. in the present paper our objective is to prove the following result: let   r be a prime k-algebra over a commutative ring   k with unity and let 1 2 ( , , , ) n f x x x be a multilinear polynomial over k. suppose that   d is a nonzero derivation on r such that ...

Let Γ be an additive group. For S ⊆ Γ, 0 6∈ S and S = {−s : s ∈ S} = S, the Cayley graph G = C(Γ, S) is the undirected graph having vertex set V (G) = Γ and edge set E(G) = {(a, b) : a − b ∈ S}. The Cayley graph G = C(Γ, S) is regular of degree |S|. For a positive integer n > 1, the unitary Cayley graph Xn = C(Zn,Z∗n) is defined by the additive group of the ring Zn of integers modulo n and the ...