Rings with xn + x or xn − x nilpotent

Let [Formula: see text] be a ring and let [Formula: see text] be an arbitrary but fixed positive integer. We characterize those rings [Formula: see text] whose elements [Formula: see text] satisfy at least one of the relations that [Formula: see text] or [Formula: see text] is a nilpotent whenever [Formula: see text]. This extends results from the same branch obtained by Danchev [A characterization of weakly J(n)-rings, J. Math. Appl. 41 (2018) 53–61], Koşan et al. [Rings with [Formula: see text] nilpotent, J. Algebra Appl. 19 (2020)] and Abyzov and Tapkin [On rings with [Formula: see text] nilpotent, J. Algebra Appl. 21 (2022)], respectively.

Multiplicative functions satisfying the functional equation κf(m^2+n^2) = f(κm^2) + f(κn^2)

For a fixed positive integer \kappa, the functional equation \kappa f(m^2 + n^2) = f(\kappa m^2) + \kappa f(n^2), (m,n\in\mathbb{N}) is solved for multiplicative functions f. This complements a 1996 result of Chung [2] which deals with the case \kappa=1. The method used relies on the sum of two squares theorem in number theory.

On new arithmetic function relative to a fixed positive integer. Part 1

The main purpose of this note is to define a new arithmetic function relative to a fixed positive integer and to study some of its properties.

Sums of Averages of GCD-Sum Functions II

Let $$ \gcd (k,j) $$ gcd ( k , j ) denote the greatest common divisor of the integers k and j, and let r be any fixed positive integer. Define $$\begin{aligned} M_r(x; f) := \sum _{k\le x}\frac{1}{k^{r+1}}\sum _{j=1}^{k}j^{r}f(\gcd (j,k)) \end{aligned}$$ M r ( x ; f ) : = ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r f ( gcd ( j , k ) ) for any large real number $$x\ge 5$$ x ≥ 5 , where f is any arithmetical function. Let $$\phi $$ ϕ , and $$\psi $$ ψ denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of $$M_r(x; \mathrm{id})$$ M r ( x ; id ) , $$M_r(x;{\phi })$$ M r ( x ; ϕ ) and $$M_r(x;{\psi })$$ M r ( x ; ψ ) . Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of $$M_r(x;\mathrm{id})$$ M r ( x ; id ) for any large positive number $$x>5$$ x > 5 satisfying $$x=[x]+\frac{1}{2}$$ x = [ x ] + 1 2 .

On the generalized Ramanujan-Nagell equation $ x^2+(2k-1)^y = k^z $ with $ k\equiv 3 $ (mod 4)

<abstract><p>Let $ k $ be a fixed positive integer with $ k &gt; 1 $. In 2014, N. Terai ^{[<xref ref-type="bibr" rid="b6">6</xref>]} conjectured that the equation $ x^2+(2k-1)^y = k^z $ has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. This is still an unsolved problem as yet. For any positive integer $ n $, let $ Q(n) $ denote the squarefree part of $ n $. In this paper, using some elementary methods, we prove that if $ k\equiv 3 $ (mod 4) and $ Q(k-1)\ge 2.11 $ log $ k $, then the equation has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. It can thus be seen that Terai's conjecture is true for almost all positive integers $ k $ with $ k\equiv 3 $(mod 4).</p></abstract>

Rank 2 preservers on symmetric matrices with zero trace

Let F be a field, V1 and V2 be vector spaces of matrices over F and let ρ be the rank function. If T :V1 → V2 is a linear map, and k a fixed positive integer, we say that T is a rank k preserver if for any matrix Aϵ, V1 ρ(A) = k implies ρ(T( A))= k . In this paper, we characterize those rank 2 preservers on symmetric matrices with zero trace under certain conditions.

Hyperstability of the k -Cubic Functional Equation in Non-Archimedean Banach Spaces

Using the fixed point approach, we investigate a general hyperstability results for the following k -cubic functional equations f k x + y + f k x − y = k f x + y + k f x − y + 2 k k 2 − 1 f x , where k is a fixed positive integer ≥ 2 , in ultrametric Banach spaces.

An optimal algorithm to find minimum k-hop connected dominating set of permutation graphs

A set [Formula: see text] is said to be a [Formula: see text]-hop dominating set ([Formula: see text]-HDS) of a graph [Formula: see text] if every vertex [Formula: see text] is within [Formula: see text]-distances from at least one vertex [Formula: see text], i.e. [Formula: see text], where [Formula: see text] is a fixed positive integer. A dominating set [Formula: see text] is said to be minimum [Formula: see text]-hop connected dominating set of a graph [Formula: see text], if it is minimal as well as it is [Formula: see text]-HDS and the subgraph of G made by [Formula: see text] is connected. In this paper, we present an [Formula: see text]-time algorithm for computing a minimum [Formula: see text]-hop connected dominating set of permutation graphs with [Formula: see text] vertices.

Generalized Skew Derivations and Nilpotent Values on Lie Ideals

Let R be a prime ring of characteristic different from 2 and 3, Qr be its right Martindale quotient ring and C be its extended centroid. Suppose that F and G are generalized skew derivations of R, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Under appropriate conditions we prove that if (F(x)x – xG(x))n = 0 for all x ∈ L, then one of the following holds: (a) there exists c ∈ Qr such that F(x) = xc and G(x) = cx; (b) R satisfies s4 and there exist a, b, c ∈ Qr such that F(x) = ax + xc, G(x) = cx + xb and (a − b)2 = 0.

Additive n-commuting maps on semiprime rings

Let R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [y, x]1 = [y, x] = yx − xy for x, y ∈ Q and inductively [y, x]k = [[y, x]k−1, x] for k > 1. Suppose that f : R → Q is an additive map satisfying [f(x), x]n = 0 for all x ∈ R, where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f(x) = λx + μ(x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.