Nonzero Integer Research Articles

An Evans function for the linearised 2D Euler equations using Hill’s determinant

We study the point spectrum of the linearisation of Euler’s equation for the ideal fluid on the torus about a shear flow. By separation of variables the problem is reduced to the spectral theory of a complex Hill’s equation. Using Hill’s determinant an Evans function of the original Euler equation is constructed. The Evans function allows us to completely characterise the point spectrum of the linearisation, and to count the isolated eigenvalues with non-zero real part. We prove that the number of discrete eigenvalues of the linearised operator for a specific shear flow is exactly twice the number of non-zero integer lattice points inside the so-called unstable disc.

Stepanov and Weyl Classes of c-Almost Periodic Type Functions

As an extension of some classes of generalized almost periodic functions, in this paper we develop the notion of c-almost periodicity in the sense of Stepanov and Weyl approaches. In fact, we extend some basic results of this theory which were already demonstrated for the standard cases. In particular, we prove that every c-almost periodic function in the sense of Stepanov approach (in the sense of equi-Weyl or Weyl approaches, respectively) is also c^m-almost periodic in the sense of Stepanov approach (in the sense of equi-Weyl or Weyl approaches, respectively) for each non-zero integer number m. This study is performed for both representative cases of functions defined on the real axis and with values in a Banach space and the complex functions defined on vertical strips in the complex plane.

ON FINDING INTEGER SOLUTIONS TO THE HOMOGENEOUS CONE X^2+(K^2+2K)Y^2=(K+1)^4 Z^2

This paper aims at determining non-zero distinct integer solutions to the homogeneous cone givenby2422z)1k(y)k2k(x. A few interesting properties betweenthe solutions are presented

Some Generalized Formula For Sums of Cube

The study of integer representations as a sum of powers is still a very long standing problem. In this work, the study of integer representation as a sum of cube is introduced and investigated for non-zero distinct integer solution. Let \(a_1\), \(a_2\), \(a_3\), ... , \(a_n\) and d be any positive integers such that \(a_n\) - \(a_n\)-1= \(a_n\)-1 - \(a_n\)-2 = ... = \(a_2\) - \(a_1\) = d. This study formulates some general results for sums of n cube. In particular, this research introduces and develops the diophantine equation I =(\(a_1\)+\(a_2\)+\(a_3\)+...+\(a_n\)) L = \(a_1^3\)+\(a_2^3\)+\(a_3^3\)+...+\(a_n^3\) for some integer L. The method involves decomposing integer I into sums of n cube and determination of general representation of integer L using case by case basis.

Virtual Knots with Properties of Kishino's Knot

Satoh and Taniguchi defined an invariant of virtual knots $J_n$ for a non-zero integer $n$. It is called the $n$-writhe. The $n$-writhes give the coefficients of some polynomial invariants for virtual knots including the index polynomial, the odd writhe polynomial and the affine index polynomial. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The unknotting number by the virtualization is called the virtual unknotting number. Kishino's knot is a virtual unknotting number one knot which has the trivial $n$-writhe and the trivial Jones polynomial. In this paper, we construct infinitely many virtual knots which have the same properties as Kishino's knot.

On b-repdigits as product of consecutive of Lucas members

A b-repdigit is a positive integer that has only one distinct digit in its base b expansion, i.e. a number of the form a(bm – 1)=(b – 1), for some positive integers m ≥ 1, b ≥ 2 and 1 ≤ a ≤ b – 1. Let r; s be non-zero integers with r ≥ 1 and s ∈ {±1}, let {Un } n ≥0 be the Lucas sequence given by Un +2 = rU n+1 + sU n , with U 0 = 0 and U 1 = 1: In this paper, we give effective bounds for the solutions of the Diophantine equation where a; b; n; k and m are positive integers such that 1 ≤ a ≤ b – 1; n; k ≥ 1 and 2 ≤ b ≤ 10. Then, we explicitly solve the above Diophantine equation for the Fibonacci, Pell and balancing sequences.

Prime power residue and linear coverings of vector space over [formula omitted

Let q be an odd prime and B={bj}j=1l be a finite set of nonzero integers that does not contain a perfect qth power. We show that B has a qth power modulo every prime p≠q and not dividing ∏b∈Bb if and only if B corresponds to a linear hyperplane covering of Fqk. Here, k is the number of distinct prime factors of the q-free part of elements of B. Consequently: (i) a set B⊂Z∖{0} with cardinality less than q+1 cannot have a qth power modulo almost every prime unless it contains a perfect qth power and (ii) For every set B={bj}j=1l⊂Z∖{0} and for every (cj)j=1l∈(Fq∖{0})l the set B contains a qth power modulo every prime p≠q and not dividing ∏j=1l if and only if the set {bjcj}j=1l does so.

Chern insulator phases and spontaneous spin and valley order in a moiré lattice model for magic-angle twisted bilayer graphene

At a certain ``magic'' relative twist angle of two graphene sheets it remains a challenge to obtain a detailed description of the proliferation of correlated topological electronic phases and their filling dependence. We perform a self-consistent real-space Hartree-Fock study of an effective moir\'e lattice model to map out the preferred ordered phases as a function of Coulomb interaction strength and moir\'e flat-band filling factor. It is found that a quantum valley Hall phase, previously discovered at charge neutrality, is present at all integer fillings for sufficiently large interactions. However, except for charge neutrality, additional spontaneous spin/valley polarization is present in the ground state at nonzero integer fillings, leading to Chern insulator phases and anomalous quantum Hall effects at odd filling factors, thus constituting an example of interaction-driven nontrivial topology. At weaker interactions, all nonzero integer fillings feature metallic inhomogeneous spin/valley ordered phases which may also break additional point group symmetries of the system. We discuss these findings in the light of previous theoretical studies on and recent experimental developments related to magic-angle twisted bilayer graphene.

On the Integer Solutions of the Homogeneous Biquadratic Diophantine Equation

Abstract: The homogenous biquadratic Diophantine equation with five unknowns non-zero unique integer solutions   4 4 2 2 2 x  y  26 z  w p are found using several techniques. The unusual numbers and the solutions are found to have a few intriguing relationships. The relationships between the solutions’ recurrences are also shown.

On The Ternary Quadratic Diophantine Equation

Abstract: The non-zero unique integer solutions to the quadratic Diophantine equation with three unknowns 2 2 2 x  14 xy  y  z are examined. We derive integral solutions in four different patterns. A few intriguing relationships between the answers and a few unique polygonal integers are shown

The Generalized Lucas Primes in the Landau’s and Shanks’ Conjectures

Landau’s conjecture and Shanks’ conjecture state that there are infinitely many prime numbers of the forms x2+1 and x4+1 for some nonzero integer , respectively. In this paper, we present a technique for studying whether or not there are infinitely many prime numbers of the form x2+1 or x4+1 derived from some Lucas sequences of the first kind {Un(P,Q)} (or simply, {Un}) or the second kind {Vn(P,Q)} (or simply, {Vn}) , where P greater or equal to 1 and Q= 1 or -1. Furthermore, as applications we represent the procedure of this technique in case of x is either an integer or a Lucas number of the first or the second kind with x greater or equal to 1 and 1 less or equal to P less or equal to 20.

Quadratic forms representing large integers only

For a positive integer n, let T(n) be the set of all integers greater than or equal to n. An integral quadratic form f is called tight T(n)-universal if the set of nonzero integers that are represented by f is exactly T(n). Let t(n) be the smallest possible rank over all tight T(n)-universal quadratic forms. In this article, we find all tight T(n)-universal diagonal quadratic forms. We also prove that t(n)∈Ω(log2⁡(n))∩O(n). Explicit lower and upper bounds for t(n) will be provided for some small integer n.

Non-Extendability of Special DIO 3-Tuples Involving Nonaganal Pyramidal Number

Abstract: This paper concerns with the construction of three distinct polynomials with integer coefficients (a1, a2, a3) such that the product of any two contribution of the set subtracted to their sum and improved by a non-zero integer (or a polynomial with integer coefficients) is a perfect square and this shows the non-extendability of Special Dio Quadruple

On Finding Integer Solutions to Sextic Equation with Three Unknowns

This paper deals with the problem of finding non-zero distinct integer solutions to the non-homogeneous ternary sextic equation given by . A few interesting relations between the solutions and special numbers are exhibited.

The Non-Homogenous Biquadratic Equation with Four Unknowns

The Non-homogenous biquadratic equation with four unknowns given by is analyzed for obtaining different sets of non-zero distinct integer solutions through employing the linear transformations.

A note on the number $a^n+ b^n – dc^n$

We say that a positive integer $d$ is special number of degree $n$ if for every integer $m$, there exist nonzero integers $a,b,c$ such that $m=a^n+b^n-dc^n$. In this paper, we investigate some necessary conditions on $n$ for existing a special number of degree $n$.

Fermat's and Catalan's equations over $ M_2(\mathbb{Z}) $

<abstract><p>Let $ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\in M_2\left(\mathbb{Z}\right) $ be a given matrix such that $ bc\neq0 $ and let $ C(A) = \{B\in M_2(\mathbb{Z}): AB = BA\} $. In this paper, we give a necessary and sufficient condition for the solvability of the matrix equation $ uX^i+vY^j = wZ^k, \, i, \, j, \, k\in\mathbb{N}, \, X, \, Y, \, Z\in C(A) $, where $ u, \, v, \, w $ are given nonzero integers such that $ \gcd\left(u, \, v, \, w\right) = 1 $. From this, we get a necessary and sufficient condition for the solvability of the Fermat's matrix equation in $ C(A) $. Moreover, we show that the solvability of the Catalan's matrix equation in $ M_2\left(\mathbb{Z}\right) $ can be reduced to the solvability of the Catalan's matrix equation in $ C(A) $, and finally to the solvability of the Catalan's equation in quadratic fields.</p></abstract>

ON HOMOGENEOUS CUBIC EQUATION WITH FOUR UNKNOWNS (x³ + y³) = 7zw²

This paper concerns with the problem of obtaining non-zero distinct integer solutions to homogeneous cubic equation with four unknowns given by x³ + y³ = 7zw². A few interesting properties among the solutions are presented.

OBSERVATION ON THE BIQUADRATIC EQUATION WITH FIVE UNKNOWNS 2(x - y)(x3 + y3) + x4 - y4 = 2(z2 - w2)p2

This paper focuses on obtaining non-zero integer quintuples (x,y, z,w, p) satisfying the bi-quadratic equation with ve unknowns given by 2(x - y)(x3 + y3) + x4 - y4 = 2(z2 - w2)p2. Various patterns of solutions are obtained by reducing the given bi-quadratic equation to solvable ternary quadratic equation through employing linear transformations.

The Investigation of the Discrete Universality of L-Functions of Elliptic Curves

In the paper, we prove the discrete universality theorem in the sense of the weak convergence of probability measures in the space of analytic functions for the L-functions of elliptic curves. We consider an approximation of analytic functions by translations LE (s+imh) , where h > 0 is a fixed number, the translations of the imaginary part of the complex variable take values from some discrete set such as arithmetical progression. We suppose that the number h > 0 is chosen so that exp{2πk/h } is an rational number for some non-zero integer. The proof of discrete universality of the derivatives of L-functions of elliptic curves is based on a limit theorem in the sense of weak convergence of probability measures in the space of analytic functions.