Positive Integer Research Articles

Korselt numbers through computational algoritms

The Korselt numbers and sets were discussed for the first time in 2007. The problem can be considered as a new one with limited literature making it as a new field of research. Let N be a positive integer and α a non-zero integer. If N ≠ α and p divides N for each prime divisor p of N, then N is called an α−Korselt number (Kα-number). In this thesis, many concepts such as Korselt numbers that are related to Carmichael numbers have been studied. Korselt determined Korselt numbers by studying the converse of Fermat’s Little theorem and it can be noticed that all these numbers depend on number theory, prime numbers, divisibility and modular arithmetic. To validate the concerned theorems, an illustrated proofs were followed through detailed steps in addition to many examples are solved in order to support the correctness of these theories. It is important to say that some errors in literature were addressed by us. Consequently, we introduced proper corrections for them. Finally, many notes have been taken and directed us to build and develop a number of complicated algorithms, some of them in order to find Korselt sets for relatively large numbers in an effective way in a short time which may require a great time and need tedious effort if it is to be calculated manually

On the exponential Diophantine equation $x^2+p^mq^n=2y^p$

We study the exponential Diophantine equation $x^2+p^mq^n=2y^p$ in positive integers $x,y,m,n$, and odd primes $p$ and $q$ using primitive divisors of Lehmer sequences in combination with elementary number theory. We discuss the solvability of this equation.

On some $$\Sigma ^{B}_{0}$$-formulae generalizing counting principles over $$V^{0}$$

AbstractWe formalize various counting principles and compare their strengths over $$V^{0}$$ V 0 . In particular, we conjecture the following mutual independence between: a uniform version of modular counting principles and the pigeonhole principle for injections, a version of the oddtown theorem and modular counting principles of modulus p, where p is any natural number which is not a power of 2, and a version of Fisher’s inequality and modular counting principles. Then, we give sufficient conditions to prove them. We give a variation of the notion of PHP-tree and k-evaluation to show that any Frege proof of the pigeonhole principle for injections admitting the uniform counting principle as an axiom scheme cannot have o(n)-evaluations. As for the remaining two, we utilize well-known notions of p-tree and k-evaluation and reduce the problems to the existence of certain families of polynomials witnessing violations of the corresponding combinatorial principles with low-degree Nullstellensatz proofs from the violation of the modular counting principle in concern.

Defective acyclic colorings of planar graphs

AbstractThis paper studies two variants of defective acyclic coloring of planar graphs. For a graph and a coloring of , a 2‐colored cycle (2CC) transversal is a subset of that intersects every 2‐colored cycle. Let be a positive integer. We denote by the minimum integer such that has a proper ‐coloring which has a 2CC transversal of size , and by the minimum size of a subset of such that is acyclic ‐colorable. We prove that for any ‐vertex 3‐colorable planar graph and for any planar graph provided that . We show that these upper bounds are sharp: there are infinitely many planar graphs attaining these upper bounds. Moreover, the minimum 2CC transversal can be chosen in such a way that induces a forest. We also prove that for any planar graph and .

Equivariant embedding of finite-dimensional dynamical systems

AbstractWe prove an equivariant version of the classical Menger-Nöbeling theorem regarding topological embeddings: Whenever a group G acts on a finite-dimensional compact metric space X, a generic continuous equivariant function from X into $$([0,1]^r)^G$$ ( [ 0 , 1 ] r ) G is a topological embedding, provided that for every positive integer N the space of points in X with orbit size at most N has topological dimension strictly less than $$\frac{rN}{2}$$ rN 2 . We emphasize that the result imposes no restrictions whatsoever on the acting group G (beyond the existence of an action on a finite-dimensional space). Moreover if G is finitely generated then there exists a finite subset $$F\subset G$$ F ⊂ G so that for a generic continuous map $$h:X\rightarrow [0,1]^{r}$$ h : X → [ 0 , 1 ] r , the map $$ h^{F}:X\rightarrow ([0,1]^{r})^{F}$$ h F : X → ( [ 0 , 1 ] r ) F given by $$x\mapsto (f(gx))_{g\in F}$$ x ↦ ( f ( g x ) ) g ∈ F is an embedding. This constitutes a generalization of the Takens delay embedding theorem into the topological category.

On the Exponential Diophantine Equation 2^x-99^y=z^2

Abstract: In this work, we determine all non-negative integers solution to the exponential Diophantine equation . The mathematical process bases on several theorems in Number theory, including the modular arithmetic, Divisibility and the Division Algorithm. The solution set to the equation is .

Smoothed asymptotics: From number theory to QFT

Inspired by the method of smoothed asymptotics developed by Terence Tao, we introduce a new ultraviolet regularization scheme for loop integrals in quantum field theory which we call η regularization. This reveals a connection between the elimination of divergences in divergent series of powers and the preservation of gauge invariance in the regularization of loop integrals in quantum field theory. In particular, we note that a method for regularizing the series of natural numbers so that it converges to minus one-twelfth inspires a regularization scheme for non-Abelian gauge theories coupled to Dirac fermions that preserves the Ward identity for the vacuum polarization tensor and other higher point functions. We also comment on a possible connection to Schwinger proper time integrals. Published by the American Physical Society 2024

N-prime elements and the primality of x − α in D⟦x⟧

Let D be an integral domain and D [ ​ [ x ] ​ ] be the power series ring over D. In this paper, we study the primality of x − α in D [ ​ [ x ] ​ ] , where α ∈ D . For this purpose, we generalize the definition of a prime element as follows. For a positive integer N, a nonzero nonunit α ∈ D is called an N-prime element if for any a , b ∈ D , α N | ab implies α | a or α | b . We prove that if α is an N-prime element for some N, then x − α is a prime element in D [ ​ [ x ] ​ ] . Surprisingly, it is shown that the converse also holds when D is a PID, a valuation domain, or a Dedekind domain. In other words, when D is a PID, a valuation domain, or a Dedekind domain, a necessary and sufficient condition for x − α to be a prime element in D [ ​ [ x ] ​ ] is α is an N-prime element in D for some N. This however does not hold for other types of integral domains such as UFDs or Krull domains. We also investigate the N-prime property in an arbitrary integral domain and give other (equivalent) conditions for an element α in the aforementioned types of integral domains to be an N-prime element.

The continuous-time magnetic quantum walk and its probability invariance on a class of graphs

Let [Formula: see text] be a nonnegative integer and [Formula: see text] the power set of the set [Formula: see text]. Then there is an adjacency relation in [Formula: see text] such that [Formula: see text] together with the relation forms a regular graph. In this paper, we propose a model of continuous-time magnetic quantum walk (MQW) on the graph [Formula: see text], and investigate its properties from a viewpoint of probability and quantum information. We first introduce a magnetic Laplacian [Formula: see text] on the graph [Formula: see text] and examine its spectrum. And then, with [Formula: see text] as the Hamiltonian, we construct our model of continuous-time MQW on the graph [Formula: see text]. We find that the model has probability distributions that are completely independent of the magnetic potential at all times. And we show that it has perfect state transfer at time [Formula: see text] when the magnetic potential satisfies some mild conditions. Some other interesting results are also obtained.

On a class of constacyclic codes of length 4ps over 𝔽pm[u] 〈u3〉

Let [Formula: see text] be a prime such that [Formula: see text], where [Formula: see text] is a positive integer. For any nonzero element [Formula: see text] of [Formula: see text], we determine the algebraic structure of all [Formula: see text]-constacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text], where [Formula: see text] and [Formula: see text] is a positive integer. If the unit [Formula: see text] is a square, [Formula: see text], each [Formula: see text]-constacyclic code of length [Formula: see text] is expressed as a direct sum of an [Formula: see text]-constacyclic code and an [Formula: see text]- constacyclic code of length [Formula: see text]. In the main case that the unit [Formula: see text] is not a square, it is shown that any nonzero polynomial of degree at most [Formula: see text] over [Formula: see text] is invertible in the ambient ring [Formula: see text]. It is also proven that the ambient ring [Formula: see text] is a local ring with the unique maximal ideal [Formula: see text], where [Formula: see text]. Such [Formula: see text]-constacyclic codes are then classified into eight distinct types of ideals, and the detailed structures of ideals in each type are provided. Among other results, the number of codewords, and the dual of each [Formula: see text]-constacyclic code are obtained. The non-existence of self-dual and isodual [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text], when the unit [Formula: see text] is not a square, is likewise proved.

Irreducible Z + -modules over a class of domains

Let d 1 , … , d r be pairwise relatively prime positive square-free integers, with d j ≥ 2 for all 1 ≤ j ≤ r . By using the elementary matrix methods, we give a complete classification of the irreducible Z + -modules over the domain Z [ d 1 , … , d r ] . Moreover, all of these irreducible Z + -modules are explicitly constructed.

On quasinormal Fitting classes

Let Zp be a group of order p and G ≀ Z p the regular wreath product of the group G with Zp . A Fitting class F is said to be X -quasinormal if there exists a natural number m such that G m ≀ Z p ∈ F whenever F ⊆ X , p is a prime, G ∈ F and G ≀ Z p ∈ X . In this paper, we generalize the well known theorem of Blessenohl and Gaschütz and prove that the intersection of any set of nonidentity X -quasinormal Fitting classes is a nonidentity X -quasinormal Fitting class. For a Fitting class F , we also discuss the relation between its X -normality and its X -quasinormality, and give the condition of X * = X ⊛ , where “ * ” is Lockett star operation and X ⊛ is the smallest nonidentity X -quasinormal Fitting class.

A Goldbach Theorem for Laurent Polynomials with Positive Integer Coefficients

We establish an analogue of the Goldbach conjecture for Laurent polynomials with positive integer coefficients.

A Goldbach Theorem for Laurent Polynomials with Positive Integer Coefficients

We establish an analogue of the Goldbach conjecture for Laurent polynomials with positive integer coefficients.

On groups with BFC-covered word values

For a group G and a positive integer n write Bn(G)={x∈G:|xG|≤n}. If s≥1 and w is a group word, say that G satisfies the (n,s)-covering condition with respect to the word w if there exists a subset S⊆G such that |S|≤s and all w-values of G are contained in Bn(G)S. In a natural way, this condition emerged in the study of probabilistically nilpotent groups of class two. In this paper we obtain the following results.Let w be a multilinear commutator word on k variables and let G be a group satisfying the(n,s)-covering condition with respect to the word w. Then G has a soluble subgroup T such that[G:T]and the derived length of T are both(k,n,s)-bounded. (Theorem 1.1.)Letk≥1and G be a group satisfying the(n,s)-covering condition with respect to the wordγk. Then (1)γ2k−1(G)has a subgroup T such that[γ2k−1(G):T]and|T′|are both(k,n,s)-bounded; and (2) G has a nilpotent subgroup U such that[G:U]and the nilpotency class of U are both(k,n,s)-bounded. (Theorem 1.2.)

Implications of Ramsey Choice principles in ZF$\mathsf {ZF}$

AbstractThe Ramsey Choice principle for families of ‐element sets, denoted , states that every infinite set has an infinite subset with a choice function on . We investigate for which positive integers and the implication is provable in . It will turn out that beside the trivial implications , under the assumption that every odd integer is the sum of three primes (known as ternary Goldbach conjecture), the only non‐trivial implication which is provable in is .

A Conjecture for the Clique Number of Graphs Associated with Symmetric Numerical Semigroups of Arbitrary Multiplicity and Embedding Dimension

A subset S of non-negative integers No is called a numerical semigroup if it is a submonoid of No and has a finite complement in No. An undirected graph G(S) associated with S is a graph having V(G(S))={vi:i∈No∖S} and E(G(S))={vivj⇔i+j∈S}. In this article, we propose a conjecture for the clique number of graphs associated with a symmetric family of numerical semigroups of arbitrary multiplicity and embedding dimension. Furthermore, we prove this conjecture for the case of arbitrary multiplicity and embedding dimension 7.

A Conjecture for the Clique Number of Graphs Associated with Symmetric Numerical Semigroups of Arbitrary Multiplicity and Embedding Dimension

A subset S of non-negative integers No is called a numerical semigroup if it is a submonoid of No and has a finite complement in No. An undirected graph G(S) associated with S is a graph having V(G(S))={vi:i∈No∖S} and E(G(S))={vivj⇔i+j∈S}. In this article, we propose a conjecture for the clique number of graphs associated with a symmetric family of numerical semigroups of arbitrary multiplicity and embedding dimension. Furthermore, we prove this conjecture for the case of arbitrary multiplicity and embedding dimension 7.

Frobenius Numbers Associated with Diophantine Triples of x2-y2=zr

We give an explicit formula for the p-Frobenius number of triples associated with Diophantine Equations x2−y2=zr (r≥2), that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of Diophantine equations x2−y2=zr. This result is also a generalization because if r=2 and p=0, the (0-)Frobenius number of the Pythagorean triple has already been given. To find p-Frobenius numbers, we use geometrically easy to understand figures of the elements of the p-Apéry set, which exhibits symmetric appearances.

Maldistributed sequences in metric spaces

Let a kind of measure μ be given on the set of all positive integers N. By a μ-maldistributed sequence in a metric space (X,d) we mean any sequence possessing μ-extremely many points in each nonempty open subspace of (X,d). Let s(X) denote the metric space of all sequences in (X,d) endowed with the Fréchet metric. In this paper we study topological properties of the set of all maldistributed sequences in X as a subspace s(X). We show that if (X,d) is separable then this set is residual for a large class of commonly used measures μ on N.