Integer Research Articles

On the Tower Factorization of Integers

Under the fundamental theorem of arithmetic, any integer n > 1 can be uniquely written as a product of prime powers pa ; factoring each exponent a as a product of prime powers qb , and so on, one will obtain what is called the tower factorization of n. Here, given an integer n > 1, we study its height h(n), that is, the number of “floors” in its tower factorization. In particular, given a fixed integer k ≥ 1 , we provide a formula for the density of the set of integers n with h(n) = k. This allows us to estimate the number of floors that a positive integer will have on average. We also show that there exist arbitrarily long sequences of consecutive integers with arbitrarily large heights.

Testability of positive integers

Given a set [Formula: see text] of some positive integers, a positive integer [Formula: see text] is [Formula: see text]-testable if there are two disjoint subsets [Formula: see text] and [Formula: see text] of [Formula: see text] such that [Formula: see text]. The testability of the set [Formula: see text] of positive integers not exceeding [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that there is a set [Formula: see text] with [Formula: see text] positive integers satisfying that each [Formula: see text] is [Formula: see text]-testable. In this short paper, we prove that [Formula: see text] and propose an algorithm to output the representation of an arbitrary positive integer by the form [Formula: see text], where [Formula: see text] and [Formula: see text] are two disjoint sets of some non-negative integers not exceeding [Formula: see text].

Digraphs of power maps over finite nilpotent groups

Given a finite group G and an integer t, let G(φt/G) be the directed graph with vertex set G and directed edges g→gt,g∈G. We prove that nilpotent groups are precisely the class of groups yielding a special symmetry on the graphs G(φt/G) that has appeared several times in the literature (considering functions over various algebraic structures). We then explore the question on when two finite nilpotent groups yield the same digraphs as t runs over Z: this defines an equivalence relation ∼φ. We completely describe the set of integers M in which the relation ∼φ is equivalent to the isomorphism of groups when one restricts to the set of nilpotent groups of order M. Our proofs rely on arguments from number theory and combinatorics.

Weak Mean Random Attractor of Reversible Selkov Lattice Systems Driven By Locally Lipschitz Lévy Noises

This paper is concerned with weak pullback mean random attractor of reversible Selkov lattice systems defined on the entire integer set \(\mathbb{Z}\) driven by locally Lipschitz Lévy noises. Firstly, we formulate the stochastic lattice equations to an abstract system defined in the non-concrete space \(\ell^2\times\ell^2\) of square-summable sequences. Secondly, we establish the global well-posedness of the systems with locally Lipschitz diffusion terms. Under certain conditions, we show that the long-time dynamics can be captured by a weakly compact and weakly attracting mean random attractor in the Bochner space \(L^2(\Omega,\ell^2\times\ell^2)\). To overcome the difficulty caused by the drift and diffusion terms, we adopt a stopping time technique to prove the convergence of solutions in probability. The mean random dynamical systems theory proposed by Wang (J. Differ. Equ., 31:2177-2204, 2019) is used to deal with the difficulty caused by the nonlinear noise.

Lattice Ordered G􀀀Semirings

Objectives: The main objective of this paper is to derive some of the results of lattice ordered semirings, distributive lattice, lattice ideals and morphisms. Methods: To establish the results, we use some conditions like commutativity, simple, multiplicative idempotent, additively idempotent, and finally, use the concept of lattice ideal in semirings. Findings: First we give some examples of lattice ordered semirings and then study some results regarding lattices, distributive lattices, commutative lattice ordered semirings and finally lattice ideals and morphisms. The unique feature of this study is that the concept of gamma is new for the study of lattices. Novelty: We consider a condition (c.f. Theorem 4.1.5) for an additively idempotent semiring due to which it becomes a distributive lattice ordered semiring. Again, in general, the sum of ideals of a semiring need not be ideal. Indeed, and are ideals of is a set of non-negative integers. Clearly, (say) is not a ideal, because , but . However, this condition does not hold in the case of a lattice ordered semiring. AMS Mathematics subject classification (2020): 16Y60. Keywords: Lattices, additive idempotent, multiplicative Γ-idempotent, k-ideal, lattice ideal, Γ-morphism

On Cilleruelo-Nathanson's method in Sidon sets

TextFor nonnegative integers h,g with h≥2, a set A of nonnegative integers is defined as a Bh[g] sequence if, for every nonnegative integer n, the number of representations of n with the form n=a1+a2+⋯+ah is no larger than g, where a1≤⋯≤ah and ai∈A for i=1,2,⋯,h. Let Z be the set of integers and N be the set of positive integers. In 2013, by introducing the method of Inserting Zeros Transformation, Cilleruelo and Nathanson obtained the following result: let f:Z→N⋃{0,∞} be any function such that lim inf|n|→∞f(n)≥g and let B be any Bh[g] sequence. Then, for any decreasing function ϵ(x)→0 as x→∞, there exists a sequence A of integers such that rA,h(n)=f(n) for all n∈Z and A(x)≫B(xϵ(x)). Recently, Nathanson further considered Sidon sets for linear forms. In this paper, we apply the Inserting Zeros Transformation into Sidon sets for linear forms and generalize the above result related to the inverse problem of representation functions. VideoFor a video summary of this paper, please visit https://youtu.be/KRtew2rLbXY.

Ratio-Covarieties of Numerical Semigroups

In this work, we will introduce the concept of ratio-covariety, as a family R of numerical semigroups that has a minimum, denoted by min(R), is closed under intersection, and if S∈R and S≠min(R), then S\{r(S)}∈R, where r(S) denotes the ratio of S. The notion of ratio-covariety will allow us to: (1) describe an algorithmic procedure to compute R; (2) prove the existence of the smallest element of R that contains a set of positive integers; and (3) talk about the smallest ratio-covariety that contains a finite set of numerical semigroups. In addition, in this paper we will apply the previous results to the study of the ratio-covariety R(F,m)={S∣S is a numerical semigroup with Frobenius number F and multiplicitym}.

First-principles study on rare earth-based equiatomic quaternary Heusler alloys YbCoCrSb and YbCoTiSn: New candidates for spintronics

We have investigated the structural, mechanical, thermal, electronic, magnetic, and half-metallic properties of new Yb-based EQHAs such as YbCoCrSb and YbCoTiSn using first-principles calculations. We computed the ground state properties of different possible crystal structures and magnetic states. We also calculated their elastic constants and derived mechanical parameters, including bulk, shear, and Young's moduli, as well as Poisson's and Pugh ratios. The electronic structure of both EQHAs shows half-metallic behavior with a spin down band gap of 0.39 eV for YbCoCrSb and 0.30 eV for YbCoTiSn in the GGA-PBE scheme. The estimated total spin magnetic moment has a positive integer value, which is well in accordance with the Slater-Pauling rule of 18. Because of the nearly closed f orbital shell of the Yb atom, the total spin magnetic moment is mainly dependent on 3d transition metal atoms. These alloys exhibit TC > 300 K using the mean field approximation.

How to solve a binary cubic equation in integers

AbstractGiven any polynomial in two variables of degree at most three with rational integer coefficients, we obtain a new search bound to decide effectively if it has a zero with rational integer coefficients. On the way we encounter a natural problem of estimating singular points. We solve it using elementary invariant theory but an optimal solution would seem to be far from easy even using the full power of the standard Height Machine.

First‐principles studies on physical properties for new half‐metallic perovskites AFeO3 (A = Ca, Sr, Ba): Spintronics and energy harvesting applications

AbstractIn this manuscript, structural, electronic, magnetic, and thermoelectric aspects of AFeO3 (A = Ca, Sr, Ba) have been calculated by means of the Full‐Potential Linearized Augmented Plane Wave Method (FP‐LAPW) using spin‐polarized Density Functional Theory (DFT). The calculated structural parameters have been found to be in good resemblance with previously available work. Enthalpy of formation and cohesive energy along with tolerance factor confirms structural stability. Electronic properties by Trans Blaha modified Becke–Johnson potential (TB‐mBJ) suggest metallic behavior in the spin‐up channel and semi‐conducting behavior in the spin‐down channel that supports half‐metallic behavior with mixed covalent and ionic bonding. Density of States (DOS) analysis confirms the major contribution of Fe‐3d‐states in the conduction band and O‐2p states in the valence band. The half‐metallic nature is further confirmed by the integer value of the total magnetic moment. The real and imaginary parts of dielectric functions, optical conductivities, absorption coefficients, reflectivity, and energy loss function have been calculated to evaluate suitability for optical applications. Thermoelectric properties with temperature range 300–900 K against chemical potential including figure of merit, Seebeck coefficient, thermal conductivities, and power factor, were examined using BoltzTraP code. Findings suggest that AFeO3 (A = Ca, Sr, Ba) compounds suitable for spintronic, thermoelectric as well as energy harvesting applications.

Near universal property of a quadratic form of Hessian discriminant 4

A quadratic form is universal if it represents all totally positive integers in a number field. In classical number theory over the rational integers there are some results where a quadratic form takes on all positive integer values except for a certain restricted set. Here we show that one of the quaternary quadratic forms proven universal for ℚ(√5) takes on all totally positive values in ℚ(√2) except for a certain infinite set. The exceptional integers have even norm and the highest power of 2 dividing that norm is 2 to the first power.

ALGEBRAIC STRUCTURES ON A SET OF DISCRETE DYNAMICAL SYSTEM AND A SET OF PROFILE

A discrete dynamical system is represented as a directed graph with graph nodes called states that can be seen on the dynamical map. This discrete dynamical system is symbolized by , where is a finite set of states and the function g is a function from to . In the dynamical map, the discrete dynamical system has a height where the number of states in each height is called a profile. The set of discrete dynamical systems has an addition operation defined as a disjoint union on the graph and a multiplication operation defined as a tensor product on the graph. The set of discrete dynamical systems and the set of profiles are very interesting to observe from the algebraic point of view. Considering operation on the set of discrete dynamical systems and the set of profiles, we can see their algebraic structure. By recognizing the algebraic structure, it will be easy to solve the polynomial equation in the discrete dynamical system and in the profile. In this research, we will investigate the algebraic structure of discrete dynamical systems and the set of profiles. This research shows that the set of discrete dynamical system has an algebraic structure, which is a commutative semiring and the set of profiles has an algebraic structure, which is a commutative semiring and -semimodule. Moreover, both sets have the same property, which is isomorphic to the set of non-negative integers.

First-principles study of structural, electronic and magnetic properties of diluted magnetic semiconductor and superlattices based‏ on Cr-doped ZnS and ZnSe compounds in wurtzite-type crystal

We performed first-principle calculations to investigate the structural, electronic, and magnetic properties of ZnS and ZnSe binary compounds, Zn0.5Cr0.5S and Zn0.5Cr0.5Se DMS alloys and (ZnS)2/Zn0.5Cr0.5Se and (ZnSe)2/Zn0.5Cr0.5S superlattices in the wurtzite structure using the full potential linear muffin–tin orbital (FP-LMTO) method. Features such as lattice constant, modulus of compressibility and its first derivative, spin-polarized band structures, total and local or partial electronic densities of states and magnetic properties were calculated. The electronic structure shows that Zn0.5Cr0.5S and Zn0.5Cr0.5Se DMS alloys and (ZnS)2/Zn0.5Cr0.5Se and (ZnSe)2/Zn0.5Cr0.5S superlattices are half-metallic ferromagnetic with 100% complete spin polarization. The total magnetic moments calculated show the same integer value of 4 µB, which confirms the ferromagnetic half-metallic behavior of these compounds. We found that the ferromagnetic state is stabilized by the p-d exchange associated with the double-exchange mechanism. Zn0.5Cr0.5S and Zn0.5Cr0.5Se DMS alloys and (ZnS)2/Zn0.5Cr0.5Se and (ZnSe)2/Zn0.5Cr0.5S superlattices are shown to be promising new candidates for applications in the fields of spintronics.

Analytic structure of the associated Legendre functions of the second kind

We consider the complex ν plane structure of the associated Legendre functions of the second kind Qν−1/2−K(cosh⁡ρ). We find that for any noninteger value of K the Qν−1/2−K(cosh⁡ρ) have an infinite number of poles in the complex ν plane, but for any negative integer K there are no poles at all. For K = 0 or any positive integer K there is only a finite number of poles, with there only being one single pole (at ν = 0) when K = 0. This pattern is analogous to the pattern of exceptional points that appear in a wide variety of physical contexts. However, while theories with exceptional points usually lose a finite number of degrees of freedom at the exceptional points, the Qν−1/2−K(cosh⁡ρ) lose an infinite number of poles whenever K is integer. Moreover, while theories with exceptional points usually have a finite number of such exceptional points, the Qν−1/2−K(cosh⁡ρ) possess an infinite number of points (all integer K) at which they lose degrees of freedom. Other than in the PT-symmetry Jordan-block case, exceptional points usually occur at complex values of parameters. While not being Jordan-block exceptional points themselves, the poles associated with the Qν−1/2−K(cosh⁡ρ) nonetheless occur at real values of K.

Invariant set generated by a nonreal number is everywhere dense

A set of complex numbers $S$ is called invariant if it is closed under addition and multiplication, namely, for any $x, y \in S$ we have $x+y \in S$ and $xy \in S$ . For each $s \in {\mathbb {C}}$ the smallest invariant set ${\mathbb {N}}[s]$ containing $s$ consists of all possible sums $\sum _{i \in I} a_i s^i$ , where $I$ runs over all finite nonempty subsets of the set of positive integers ${\mathbb {N}}$ and $a_i \in {\mathbb {N}}$ for each $i \in I$ . In this paper, we prove that for $s \in {\mathbb {C}}$ the set ${\mathbb {N}}[s]$ is everywhere dense in ${\mathbb {C}}$ if and only if $s \notin {\mathbb {R}}$ and $s$ is not a quadratic algebraic integer. More precisely, we show that if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is a transcendental number, then there is a positive integer $n$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for either $t=s$ or $t=s+s^2$ . Similarly, if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is an algebraic number of degree $d \ne 2, 4$ , then there are positive integers $n, m$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for $t=ms+s^2$ . For quadratic and some special quartic algebraic numbers $s$ it is shown that a similar sumset of three sets cannot be dense. In each of these two cases the density of ${\mathbb {N}}[s]$ in ${\mathbb {C}}$ is established by a different method: for those special quartic numbers, it is possible to take a sumset of four sets.

On the Vertex Identification Spectra of Grids

Let [Formula: see text] be a red–white coloring of the vertices of a nontrivial connected graph [Formula: see text] with diameter [Formula: see text], where at least one vertex is colored red. Then [Formula: see text] is called an identification coloring or simply, an ID-coloring, if and only if for any two vertices [Formula: see text] and [Formula: see text], [Formula: see text], where for any vertex [Formula: see text], [Formula: see text] and [Formula: see text] is the number of red vertices at a distance [Formula: see text] from [Formula: see text]. A graph is said to be an ID-graph if it possesses an ID-coloring. If [Formula: see text] is an ID-graph, then the spectrum of [Formula: see text] is the set of all positive integers [Formula: see text] for which [Formula: see text] has an ID-coloring with [Formula: see text] red vertices. The identification number or ID-number of a graph is the smallest element in its spectrum. In this paper, we extend a result of Kono and Zhang on the identification number of grids [Formula: see text]. In particular, we give a formulation of strong ID-coloring and use it to give a sufficient condition for an ID-coloring of a graph to be extendable to an ID-coloring of the Cartesian product of a path [Formula: see text] with [Formula: see text]. Consequently, some elements of the spectrum of grids [Formula: see text] for positive integers [Formula: see text] and [Formula: see text], with [Formula: see text], are obtained. The complete spectrum of ladders [Formula: see text] is then determined using systematic constructions of ID-colorings of the ladders.

On the x-coordinates of Pell equations that are products of two Pell numbers

Abstract Let (Pm ) m≥0 be the sequence of Pell numbers given by P 0 = 0, P 1 = 1, and P m+2 = 2P m+1 + Pm for all m ≥ 0. In this paper, for an integer d ≥ 2 which is square free, we show that there is at most one value of the positive integer x participating in the Pell equation x 2 − dy 2 = ± 1, which is a product of two Pell numbers.

Numerically explicit estimates for the distribution of rough numbers

For x≥y>1 and u:=log⁡x/log⁡y, let Φ(x,y) denote the number of positive integers up to x free of prime divisors less than or equal to y. In 1950 de Bruijn [4] studied the approximation of Φ(x,y) by the quantityμy(u)eγxlog⁡y∏p≤y(1−1p), where γ=0.5772156... is Euler's constant andμy(u):=∫1uyt−uω(t)dt. He showed that the asymptotic formulaΦ(x,y)=μy(u)eγxlog⁡y∏p≤y(1−1p)+O(xR(y)log⁡y) holds uniformly for all x≥y≥2, where R(y) is a positive decreasing function related to the error estimates in the Prime Number Theorem. In this paper we obtain numerically explicit versions of de Bruijn's result.

Frieze patterns over algebraic numbers

AbstractConway and Coxeter have shown that frieze patterns over positive rational integers are in bijection with triangulations of polygons. An investigation of frieze patterns over other subsets of the complex numbers has recently been initiated by Jørgensen and the first two authors. In this paper, we first show that a ring of algebraic numbers has finitely many units if and only if it is an order in a quadratic number field where . We conclude that these are exactly the rings of algebraic numbers over which there are finitely many non‐zero frieze patterns for any given height. We then show that apart from the cases all non‐zero frieze patterns over the rings of integers for have only integral entries and hence are known as (twisted) Conway–Coxeter frieze patterns.

Distance magic labeling of join of graphs and big data analysis

Consider a simple graph G = 〈V, E〉 with n vertices. Define a function f from vertex set of G to set of integers 1, 2, …, n subject to the condition that there exists an integer k > 0 such that the sum of adjacent labels of each vertex in G is equal to k. In this paper, we prove that the graph K n - ⌊ n 2 ⌋ e has DML for all n ≥ 1, decomposition of distance magic graph K2n - {ne} into n ( n - 1 ) 2 edge distinct copies of C4 for all n ≥ 2 and DML of join of complete graphs gives unique mathematical model and it will be a useful model in big data analysis.