Prime Integers Research Articles

Castelnuovo–Mumford Regularity of Projective Monomial Curves via Sumsets

Let A={a_0,ldots ,a_{n-1}} be a finite set of nge 4 non-negative relatively prime integers, such that 0=a_0<a_1<cdots <a_{n-1}=d. The s-fold sumset of A is the set sA of integers that contains all the sums of s elements in A. On the other hand, given an infinite field k, one can associate with A the projective monomial curve mathcal {C}_A parametrized by A, CA={(vd:ua1vd-a1:⋯:uan-2vd-an-2:ud)∣(u:v)∈Pk1}⊂Pkn-1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\quad \\mathcal {C}_A=\\{(v^d:u^{a_1}v^{d-a_1}:\\cdots :u^{a_{n-2}}v^{d-a_{n-2}}:u^d) \\mid (u:v)\\in \\mathbb {P}^{1}_k\\}\\subset \\mathbb {P}^{n-1}_k. \\end{aligned}$$\\end{document}The exponents in the previous parametrization of mathcal {C}_A define a homogeneous semigroup mathcal {S}subset mathbb {N}^2. We provide several results relating the Castelnuovo–Mumford regularity of mathcal {C}_A to the behavior of the sumsets of A and to the combinatorics of the semigroup mathcal {S} that reveal a new interplay between commutative algebra and additive number theory.

Laplacian eigenvalues of the unit graph of the ring [formula omitted

We investigate the Laplacian eigenvalues about unit graph G(Zn) on Zn and show the case whenever n is an even number or an odd prime power, G(Zn) would be Laplacian integral. We also prove that if n>1, then the Laplacian spectral radius of G(Zn) is equal cardinal number of V(G(Zn)) if and only if n=pk, here p is a prime integer, k is an integer that positive. In addition, this paper also characterizes n that algebraic connectivity of G(Zn) coincides with the vertex connectivity.

The 2-Class Group of Certain Families of Imaginary Triquadratic Fields

ABSTRACT Our goal in this paper is to compute the 2-rank of the class group of the fields K = ℚ ( − 2 , q , d ) \[\mathbb{K}=\mathbb{Q}\left( \sqrt{-2},\sqrt{q},\sqrt{d} \right)\] for any odd positive squarefree integer d and any prime integer q ≡ 5 (mod 8). Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic, of type (2, 2), (2, 4) or (2, 2, 2).

Black Lenses in Kaluza-Klein Matter.

We present the first examples of formally asymptotically flat black hole solutions with horizons of general lens space topology L(p,q). These five-dimensional static or stationary spacetimes are regular on and outside the event horizon for any choice of relatively prime integers 1≤q<p; in particular, conical singularities are absent. They are supported by Kaluza-Klein matter fields arising from higher dimensional vacuum solutions through reduction on tori. The technique is sufficiently robust that it leads to the explicit construction of regular solutions, in any dimension, realizing the full range of possible topologies for the horizon as well as the domain of outer communication, that are allowable with multi-axisymmetry. Lastly, as a by-product, we obtain new examples of regular gravitational instantons in higher dimensions.

Radon transform on algebraic Cormack α-curves inversion formula

Radon transforms (RTs) in , defined on simple integration manifolds such as straight lines or circles, have attracted extensive interest since they provide the reconstruction of physical density functions in various applicative inverse problems, via a closed form inversion formula. In an attempt to see whether RT on more involved integration manifolds would still admit closed form inversion formulas, A M Cormack has sought to invert RT defined on a class of non-trivial curves in the plane, called α-curves. They are generated by application of a specially conceived geometric transform on straight lines, so that the RT inversion process remains as close as possible to that of the RT on the lines. It was hoped that the inversion of RT on lines would show the way to invert the RT on α-curves. However Cormack could only establish a closed form inversion formula for RT on —curves, with . In this paper, we generalize his formula to RT on algebraic α-curves, i.e. for with , a pair of positive relatively prime integers and thereby give support to his initial conjecture.

An algebraic construction technique for codes over Hurwitz integers

Let $\alpha$ be a prime Hurwitz integer. $\mathcal{H}_{\alpha}$, which is the set of residual class with respect to related modulo function in the rings of Hurwitz integers, is a subset of $\mathcal{H},$ which is the set of all Hurwitz integers. In this study, we present an algebraic construction technique, which is a modulo function formed depending on two modulo operations, for codes over Hurwitz integers. We consider left congruent modulo $\alpha,$ and the domain of related modulo function is $\mathbb{Z}_{N(\alpha)},$ which is residual class ring of ordinary integers with $N(\alpha)$ elements. Therefore, we obtain the residue class rings of Hurwitz integers with $N(\alpha)$ size. In addition, we present some results for mathematical notations used in two modulo functions, and for the algebraic construction technique formed depending upon two modulo functions. Moreover, we presented graphs obtained by graph layout methods, such as spring, high-dimensional, and spiral embedding, for the set of the residual class obtained with respect to the related modulo function in the rings of Hurwitz integers.

On the spectral theory of groups of automorphisms of S-adic nilmanifolds

AbstractLet $S=\{p_1, \ldots , p_r,\infty \}$ for prime integers $p_1, \ldots , p_r.$ Let X be an S-adic compact nilmanifold, equipped with the unique translation-invariant probability measure $\mu .$ We characterize the countable groups $\Gamma $ of automorphisms of X for which the Koopman representation $\kappa $ on $L^2(X,\mu )$ has a spectral gap. More specifically, let Y be the maximal quotient solenoid of X (thus, Y is a finite-dimensional, connected, compact abelian group). We show that $\kappa $ does not have a spectral gap if and only if there exists a $\Gamma $ -invariant proper subsolenoid of Y on which $\Gamma $ acts as a virtually abelian group,

Prime factorization via localized tile assembly in a DNA origami framework

Modern cybersecurity built on public-key cryptosystems like Rivest-Shamir-Adleman is compromised upon finding solutions to the prime factorization. Nevertheless, solving the prime factorization problem, given a large N, remains computationally challenging. Here, we design DNA origami frameworks (DOFs) to direct localized assembly of double-crossover (DX) tiles for solving prime factorization with a model consisting of the computing, decision-making, and reporting motifs. The model implementation is based on the sequential assembly of different DX tiles in the DOF cavity that carries overhangs encoding the prime and composite integers. The primes are multiplied and then verified with the composite, and the result is visualized under atomic force microscopy via the presence (success) or absence (failure) of biotin-streptavidin labels on the reporting DX tile. The factorization of semiprimes 6 and 15 is realized with this DOF-based demonstration. Given the potential of massively parallel processing ability of DNA, this strategy opens an avenue to solve complex mathematical puzzles like prime factoring with molecular computing.

Sum Graph of Z_(p^n q^m ) Groups and Some Topological Indices of G+ (Z_(p^n q^m ))

This study introduces the extended graph theory in the sum group via , where p and q are prime integers, and the sum is more than the order of the group . If the group sum graph of it meets the properties of the graph theory and computes all the degrees of graphs, and it will be linked, cyclic, etc. will also use a generalized version of it to calculate the well-known topological indices.

Polycyclic codes over [formula omitted]: Classification, Hamming distance, and annihilators

Let p be prime integer, s and m positive integers; f(x) be a monic irreducible polynomial over R where R:=Fpm[u]/〈u2〉. This paper gives a complete classification of polycyclic codes over R associated to f(x)ps, and the other hand, when f(x)∈Fpm[x], their Hamming distances and their annihilators are explicitly given.

Acceleration of Wheel Factoring Techniques

The efficiency with which an integer may be factored into its prime factors determines several public key cryptosystems’ security in use today. Although there is a quantum-based technique with a polynomial time for integer factoring, on a traditional computer, there is no polynomial time algorithm. We investigate how to enhance the wheel factoring technique in this paper. Current wheel factorization algorithms rely on a very restricted set of prime integers as a base. In this study, we intend to adapt this notion to rely on a greater number of prime integers, resulting in a considerable improvement in the execution time. The experiments on composite numbers n reveal that the proposed algorithm improves on the existing wheel factoring algorithm by about 75%.

A comparison of dg algebra resolutions with prime residual characteristic

Abstract In this article, we fix a prime integer p and compare certain dg algebra resolutions over a local ring whose residue field has characteristic p. Namely, we show that given a closed surjective map between such algebras there is a precise description for the minimal model in terms of the acyclic closure and that the latter is a quotient of the former. A first application is that the homotopy Lie algebra of a closed surjective map is abelian. We also use these calculations to show deviations enjoy rigidity properties which detect the (quasi-)complete intersection property.

FERMAT’S LAST THEOREM IS EQUIVALENT TOBEAL’S CONJECTURE

It is proved in this paper that (1) Fermat’s Last Theorem: If π is an odd prime, there are no relatively prime positive integers x,y,z satisfying the equation zπ = xπ+yπ, and (2) Beal’s Conjecture :The equation zξ = xµ +yν has no solution in relatively prime positive integers x,y,z with µ,ξ and ν odd primes at least 3. It is also proved that these two statements, (1) and (2), are equivalent.

On distribution of the number of semisimple rings of order at most x in an arithmetic progression

Let \ell and q denote relatively prime positive integers. In this article, we derive the asymptotic formula for the summation \begin{align*} \sum_{n\leq x\atop n\equiv \ell \pmod q}S(n), \end{align*} where S(n) denotes the number of non-isomorphic finite semisimple rings with n elements.

The Frobenius problem for special progressions

&lt;abstract&gt;&lt;p&gt;Let $ S $ be a given finite set of positive and relatively prime integers. Denote $ L(S) $ to be the set of integers obtained by taking all nonnegative integer linear combinations of integers in $ S $. It is well known that there are finitely many positive integers that are not in $ L(S) $. Let $ g(S) $ and $ n(S) $ represent the greatest integer that does not belong to $ L(S) $ and the number of nonnegative integers that do not belong to $ L(S) $, respectively. The Frobenius problem is to determine $ g(S) $ and $ n(S) $. In 2016, Tripathi obtained results on $ g(S) $ and $ n(S) $ when $ S = \{a, ha+d, ha+db, ha+db^2, \ldots, ha+db^k\} $. In this paper, for $ S_c: = \{a, ha+d, ha+c+db, ha+2c+db^2, \ldots, ha+kc+db^k\} $ with $ h, c $ being nonnegative integers, $ a, b, d $ being positive integers and $ \gcd(a, d) = 1 $, we focused the investigation on formulas for $ g(S_c) $ and $ n(S_c) $. Actually, we gave formulas for $ g(S_c) $ and $ n(S_c) $ for all sufficiently large values of $ d $ when $ c $ is any multiple of $ d $ or certain multiples of $ a $. This generalized the results of Tripathi in 2016.&lt;/p&gt;&lt;/abstract&gt;

DETECTION OF RELATIVELY PRIME INTEGER SOLUTIONS FOR TWO DISPARATE FORMS OF MORDELL CURVES

Abstract In this paper, two unrelated natures of Mordell curves y³ = x² + k where k is a multinomial of degree four and six are scrutinized for many sets of relatively prime integer solutions. The geometrical description of the curves for each set of solutions are also exhibited with an assistance of MATLAB tools.

On a conjecture of Ramírez Alfonsín and Skałba

Let 1⩽a<b be two relatively prime integers. Sylvester found that ab−a−b is the largest integer which can not be represented by ax+by(x,y∈Z⩾0) about 160 years ago and this number shall be denoted by ga,b. Let N(a,b)={n:n⩽ga,b,n=ax+by,x,y∈Z⩾0} and πa,b be the number of primes in N(a,b). Recently, Ramírez Alfonsín and Skałba proved thatπa,b≫εga,b(log⁡ga,b)2+ε for any fixed ε>0. They further conjectured that the order of the magnitude of πa,b is 12π(ga,b), where π(x) is the number of all primes up to x. In this paper, we show that the conjecture is true for almost all pairs a,b with 1⩽a<b and (a,b)=1. The proofs rely heavily on the Bombieri–Vinogradov theorem and Brun–Titchmarsh theorem.

Slow-fast torus knots

The aim of this paper is to study global dynamics of $C^\infty$-smooth slow-fast systems on the $2$-torus of class $C^\infty$ using geometric singular perturbation theory and the notion of slow divergence integral. Given any $m\in\mathbb{N}$ and two relatively prime integers $k$ and $l$, we show that there exists a slow-fast system $Y_{\epsilon}$ on the $2$-torus that has a $2m$-link of type $(k,l)$, i.e. a (disjoint finite) union of $2m$ slow-fast limit cycles each of $(k,l)$-torus knot type, for all small $\epsilon>0$. The $(k,l)$-torus knot turns around the $2$-torus $k$ times meridionally and $l$ times longitudinally. There are exactly $m$ repelling limit cycles and $m$ attracting limit cycles. Our analysis: a) proves the case of normally hyperbolic singular knots, and b) provides sufficient evidence to conjecture a similar result in some cases where the singular knots have regular nilpotent contact with the fast foliation.

On the oscillation of certain second-order linear differential equations

AbstractThis paper consists of three parts: First, letting $b_1(z)$, $b_2(z)$, $p_1(z)$ and $p_2(z)$ be nonzero polynomials such that $p_1(z)$ and $p_2(z)$ have the same degree $k\geq 1$ and distinct leading coefficients $1$ and $\alpha$, respectively, we solve entire solutions of the Tumura–Clunie type differential equation $f^{n}+P(z,\,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$, where $n\geq 2$ is an integer, $P(z,\,f)$ is a differential polynomial in $f$ of degree $\leq n-1$ with coefficients having polynomial growth. Second, we study the oscillation of the second-order differential equation $f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}]f=0$ and prove that $\alpha =[2(m+1)-1]/[2(m+1)]$ for some integer $m\geq 0$ if this equation admits a nontrivial solution such that $\lambda (f)&lt;\infty$. This partially answers a question of Ishizaki. Finally, letting $b_2\not =0$ and $b_3$ be constants and $l$ and $s$ be relatively prime integers such that $l&gt; s\geq 1$, we prove that $l=2$ if the equation $f''-(e^{lz}+b_2e^{sz}+b_3)f=0$ admits two linearly independent solutions $f_1$ and $f_2$ such that $\max \{\lambda (f_1),\,\lambda (f_2)\}&lt;\infty$. In particular, we precisely characterize all solutions such that $\lambda (f)&lt;\infty$ when $l=2$ and $l=4$.

On the arithmetic of a family of superelliptic curves

Let p be a prime, let r and q be powers of p, and let a and b be relatively prime integers not divisible by p. Let \(C/{\mathbb {F}}_{r}(t)\) be the superelliptic curve with affine equation \(y^b+x^a=t^q-t\), and let J be the Jacobian of C. By work of Pries and Ulmer (Trans Am Math Soc 368(12):8553–8595, 2016), J satisfies the Birch and Swinnerton-Dyer conjecture (BSD). Generalizing work of Griffon and Ulmer (Pacific J Math 305(2):597–640, 2020) , we compute the L-function of J in terms of certain Gauss sums. In addition, we estimate several arithmetic invariants of J appearing in BSD, including the rank of the Mordell–Weil group \(J({\mathbb {F}}_{r}(t))\), the Faltings height of J, and the Tamagawa numbers of J in terms of the parameters a, b, q. For any p and r, we show that for certain a and b depending only on p and r, these Jacobians provide new examples of families of simple abelian varieties of fixed dimension and with unbounded analytic and algebraic rank as q varies through powers of p. Under a different set of criteria on a and b, we prove that the order of the Tate–Shafarevich group grows exponentially fast in q as \(q \rightarrow \infty \).