Ring Of Gaussian Integers Research Articles

On the Line Graph of the Complement Graph for the Ring of Gaussian Integers Modulo n

The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.

THE DIGRAPH OF THE SQUARE MAPPING ON QUOTIENT RINGS OVER THE GAUSSIAN INTEGERS

In this work, we investigate the structure of the digraph [Formula: see text] associated with the square mapping on the ring of Gaussian integers by using the exponent of the unit group modulo γ. The formula for the fixed points of [Formula: see text] is established. Some connections of the lengths of cycles with the exponent of the unit group modulo γ are presented. Furthermore, we study the maximum distance from the cycle on each component.

On the distribution of powers of a complex number

Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξαn + ν, n = 0, 1, 2, . . . , modulo \({\mathbb{Z}[i],}\) where \({i=\sqrt{-1}}\) and \({\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}}\) is the ring of Gaussian integers. For any \({z\in \mathbb{C},}\) one may naturally call the quantity z modulo \({\mathbb{Z}[i]}\)the fractional part of z and write {z} for this, in general, complex number lying in the unit square \({S:=\{z\in\mathbb{C}:0\leq \mathfrak{R}(z),\mathfrak{J}(z) 1 then there are two limit points of the sequence {ξαn +ν}, n = 0, 1, 2, . . . , which are ‘far’ from each other (in terms of α only), except when α is an algebraic integer whose conjugates over \({\mathbb{Q}(i)}\) all lie in the unit disc |z| ≤ 1 and \({\xi\in\mathbb{Q}(\alpha,i).}\) Then we prove a result in the opposite direction which implies that, for any fixed \({\alpha\in\mathbb{C}}\) of modulus greater than 1 and any sequence \({z_n\in\mathbb{C},n=0,1,2,\dots,}\) there exists \({\xi \in \mathbb{C}}\) such that the numbers ξαn−zn, n = 0, 1, 2, . . . , all lie ‘far’ from the lattice \({\mathbb{Z}[i]}\). In particular, they all can be covered by a union of small discs with centers at \({(1+i)/2+\mathbb{Z}[i]}\) if |α| is large.

Coincidence isometries of a shifted square lattice

We consider the coincidence problem for the square lattice that is translated by an arbitrary vector. General results are obtained about the set of coincidence isometries and the coincidence site lattices of a shifted square lattice by identifying the square lattice with the ring of Gaussian integers. To illustrate them, we calculate the set of coincidence isometries, as well as generating functions for the number of coincidence site lattices and coincidence isometries, for specific examples.

Polynomial extensions of IDF-domains and of IDPF-domains

An integral domain is IDF if every non-zero element has only finitely many non-associate irreducible divisors. We investigate when R IDF implies that the ring of polynomials R[T] is IDF. This is true when R is Noetherian and integrally closed, in particular when R is the coordinate ring of a non-singular variety. Some coordinate rings R of singular varieties also give R[T] IDF. Analogous results for the related concept of IDPF are also given. The main result on IDF in this paper states that every countable domain embeds in another countable domain R such that R has no irreducible elements, hence vacuously IDF, and the polynomial ring R[T] is not IDF. This resolves an open question. It is also shown that some subrings R of the ring of Gaussian integers known to be IDPF also have the property that R[T] is not IDPF.

Self-similar hex-sums of squares

Self-Similar Hex-Sums of Squares R. BREU ~ eading the very stimulating article by A. van der Poorten, et al. in the Mathematical Intelligencer [1], I realized that self-similar sums of squares have ~. ~ , a nice analogue in what I call self-similar hex-sums of squares, just as decomposing primes of the form 4q + 1 into sums of squares a 2 + b 2 o r factoring 4q + 1 in the ring of Gaussian integers has an analogue in the ring of Eisenstein integers, that is, decomposing primes of the form 3q + 1 into "hex-sums" of squares a 2 + b 2 a b . ("Hex," as the fundamental domain in the Eisenstein lattice is a hexagon, versus a square in the Gaussian case.) Eisenstein integers have the form a -+ b~0, where ~o = ( 1 + X/-Z-3)/2. Their n o r m a 2 -4b 2 + ab corresponds to the n o r m a 2 + b 2 o f Gaussian integers. Both signs can be chosen, as the "upper sign" version is just the mirrored version of the "lower sign," but then have to be used consistently throughout, that is, always the lower or always the upper sign. A norm equation of Gaussian integers, say (a 2 + b 2) (c 2 + d 2) = (ac + bd) 2 + (ad b c ) 2, translates into a similar norm equation of Eisenstein integers

Perfect Codes for Metrics Induced by Circulant Graphs

An algebraic methodology for defining new metrics over two-dimensional signal spaces is presented in this work. We have mainly considered quadrature amplitude modulation (QAM) constellations which have previously been modeled by quotient rings of Gaussian integers. The metric over these constellations, based on the distance concept in circulant graphs, is one of the main contributions of this work. A detailed analysis of some degree-four circulant graphs has allowed us to detail the weight distribution for these signal spaces. A new family of perfect codes over Gaussian integers will be defined and characterized by providing a solution to the perfect t-dominating set problem over the circulant graphs presented. Finally, we will show how this new metric can be extended to other signal sets by considering hexagonal constellations and circulant graphs of degree six.

Factor rings of the Gaussian integers

Whereas the homomorphic images of Z (the ring of integers) are well known, namely Z, {0} and Zn (the ring of integers modulo n), the same is not true for the homomorphic im-ages of Z[i] (the ring of Gaussian integers). More generally, let m be any nonzero square free integer (positive or negative), and consider the integral domain Z[ √m]={a + b √m | a, b ∈ Z}. Which rings can be homomorphic images of Z[ √m]? This ques-tion offers students an infinite number (one for each m) of investigations that require only undergraduate mathematics. It is the goal of this article to offer a guide to the in-vestigation of the possible homomorphic images of Z[ √m] using the Gaussian integers Z[i] as an example. We use the fact that Z[i] is a principal ideal domain to prove that if I =(a+bi) is a nonzero ideal of Z[i], then Z[i]/I ∼ = Zn for a positive integer n if and only if gcd{a, b} =1, in which case n = a2 + b2 . Our approach is novel in that it uses matrix techniques based on the row reduction of matrices with integer entries. By characterizing the integers n of the form n = a2 + b2 , with gcd{a, b} =1, we obtain the main result of the paper, which asserts that if n ≥ 2, then Zn is a homomorphic image of Z[i] if and only if the prime decomposition of n is 2α0 pα1 1 ··· pαk k , with α0 ∈{0, 1},pi ≡ 1(mod 4) and αi ≥ 0 for every i ≥ 1. All the fields which are homomorpic images of Z[i] are also determined.

Generalized number systems in Euclidean spaces

The concept of number systems in higher-dimensional Euclidean spaces as well as in number fields are introduced and treated. The paper is mainly a survey of known results. If ofZ k denotes the ring of integer vectorials in Euclidean space R k then we study the subgroup L= MZ k where M is a k × k matrix with integer entries. A digit set is defined with the help of residue classes ( mod L) A j ( j = 0,…, t − 1), where t = |det M|. The analogous of q-additive and q-multiplicative functions over the ring of Gaussian integers are also given.

Efficient Algorithms for gcd and Cubic Residuosity in the Ring of Eisenstein Integers

We present simple and efficient algorithms for computing gcd and cubic residuosity in the ring of Eisenstein integers, Z[zeta] , i.e. the integers extended with zeta , a complex primitive third root of unity. The algorithms are similar and may be seen as generalisations of the binary integer gcd and derived Jacobi symbol algorithms. Our algorithms take time O(n^2) for n bit input. This is an improvement from the known results based on the Euclidian algorithm, and taking time O(n· M(n)), where M(n) denotes the complexity of multiplying n bit integers. The new algorithms have applications in practical primality tests and the implementation of cryptographic protocols. The technique underlying our algorithms can be used to obtain equally fast algorithms for gcd and quartic residuosity in the ring of Gaussian integers, Z[i].

Even Unimodular Gaussian Lattices of Rank 12

We classify even unimodular Gaussian lattices of rank 12, that is, even unimodular integral lattices of rank 12 over the ring of Gaussian integers. This is equivalent to the classification of the automorphisms τ with τ2=−1 in the automorphism groups of all the Niemeier lattices, which are even unimodular (real) integral lattices of rank 24. There are 28 even unimodular Gaussian lattices of rank 12 up to equivalence.

Divisor Functions in the Ring of Gaussian Integers Weighted by the Generalized Klosterman Sum

Let τa,b,c (ω) be the number of factorizations of a Gaussian number ω in the form ω = ω1aω2bω3c, where a, b, and c are natural numbers. In the ring of Gaussian numbers, we construct an asymptotic formula for a summatory function of τa,b,c (ω) weighted by the generalized Klosterman function.

The complex sum of digits function and primes

Canonical number systems in the ring of gaussian integers ℤ[i] are the natural generalization of ordinary q-adic number systems to ℤ[i]. It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number b. In this paper we investigate the sum of digits function ν b of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the f-th power of a prime. Furthermore, we establish an Erdös-Kac type theorem for ν b . In all proofs the equidistribution of ν b in residue classes plays a crucial rôle. Starting from this fact we use sieve methods and a version of the model of Kubilius to prove our results.

FRACTILES DERIVED FROM GENERALIZED DIGIT SYSTEMS

We introduce a general method of obtaining connected replicating fractiles based on generalized digit systems in the ring of Gaussian integers.

The topological Hochschild homology of the Gaussian integers

The calculation in this paper gives the 2-torsion in the topological Hochschild homology of rings of integers in quadratic extensions of the rationals. In general, it is not hard to deduce the p -torsion in the topological Hochschild homology of rings of integers which do not ramify at p from the corresponding torsion for the integers. Thus, for the ring of Gaussian integers, and also for the integers with the square root of 2 or -2 adjoined, this paper completes the calculation of the topological Hochschild homology spectrum (which is a priori known to be a product of Eilenberg-MacLane spectra). Precisely because of this a priori knowledge, the homotopy groups can be found by doing a homology calculation. A spectral sequence arising from the filtration of topological Hochschild homology by simplicial 'skeleta' converges to the desired homology. If one reduces the rings in question modulo 2, the spectral sequence collapses at its second term. This comparison bounds the nontriviality of higher differentials in the spectral sequence for the original ring. It is explicitly demonstrated, using simplicial calculations, that the higher differentials are as nontrivial as they could be, given this bound.

Quadratic forms in normal open induction

AbstractModels of normal open induction (NOI) are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano's induction axioms for open formulas in the language of ordered semirings.Here we study the problem of representability of an element a of a model M of NOI (in some extension of M) by a quadratic form of the type X2 + b Y2 where b is a nonzero integer. Using either a trigonometric or a hyperbolic parametrization we prove that except in some trivial cases, M[x, y] with x2 + by2 = a can be embedded in a model of NOI.We also study quadratic extensions of a model M of NOI; we first prove some properties of the ring of Gaussian integers of M. Then we study the group of solutions of a Pell equation in NOI; we construct a model in which the quotient group by the squares has size continuum.

Parameter determination for complex number-theoretic transforms using cyclotomic polynomials

Some new results for finding all convenient moduli m for a complex number-theoretic transform with given transform length n and given primitive nth root of unity modulo m are presented. The main result is based on the prime factorization for values of cyclotomic polynomials in the ring of Gaussian integers.

Corrigendum: ‘‘Computing an arithmetic constant related to the ring of Gaussian integers” [Math. Comp. {\bf 44} (1985), no. 169, 241–250; MR0771043 (87a:11028)] by Gramain and M. Weber

Corrigendum: ‘‘Computing an arithmetic constant related to the ring of Gaussian integers” [Math. Comp. {\bf 44} (1985), no. 169, 241–250; MR0771043 (87a:11028)] by Gramain and M. Weber

Computing an arithmetic constant related to the ring of Gaussian integers

We compute the analogue for Z [ i ] {\mathbf {Z}}[i] of Euler’s constant, that is δ = lim n → + ∞ δ n \delta = {\lim _{n \to + \infty }}{\delta _n} , where δ n = ( Σ 2 ⩽ k ⩽ n 1 / π r k 2 ) − log ⁡ n {\delta _n} = ({\Sigma _{2 \leqslant k \leqslant n}}1/\pi r_k^2) - \log n . For this purpose we give an estimate for \[ r k = min { r ⩾ 0 ; there exists z ∈ C such that card ( Z [ i ] ∩ D ¯ ( z , r ) ) ⩾ k } , {r_k} = \min \left \{ {r \geqslant 0;{\text {there exists}}\;z \in {\mathbf {C}}\;{\text {such that card}}({\mathbf {Z}}[i] \cap \bar D(z,r)) \geqslant k} \right \}, \] and we compute a great number of values of r k {r_k} .

Computing an Arithmetic Constant Related to the Ring of Gaussian Integers

Computing an Arithmetic Constant Related to the Ring of Gaussian Integers