Results 11 to 20 of about 391,951 (282)
Formalized Mathematics, 2013 Summary Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above.
Yuichi Futa +3 more
openaire +5 more sources
On sums over Gaussian integers [PDF]
Transactions of the American Mathematical Society, 1975 The object of this paper is to give asymptotic estimates for some number theoretic sums over Gaussian integers. As a consequence of general estimates, asymptotic estimates with explicit error terms for the number of Gaussian integers with only “large” prime factors and for the number of Gaussian integers with only “small” prime factors are given.
D. G. Hazlewood
openaire +2 more sources
Gaussian integers with small prime factors
International Journal of Mathematics and Mathematical Sciences, 1979 Let ψG(xt,x) denote the number of Gaussian integers with norm not exceeding x2t whose Gaussian prime factors have norm not exceeding x2. Previous estimates have required restrictions on the parameter t with respect to x.
D. G. Hazlewood
doaj +2 more sources
Gaussian Integers and Other Quadratic Integer Rings
, 2021 This thesis deals with quadratic integer rings, in particular the Gaussian integers Z}[i]. Concepts such as quadratic extensions, Euclidean domains and unique factorization domains will be introduced to the reader. The goal of this thesis is to show how a natural generalization of the integers Z, in the form of the Gaussian integers, can be used to ...
Landin, Erik, Hussein, Seif
core +4 more sources
Polynomials which take Gaussian integer values at Gaussian integers [PDF]
Journal of Number Theory, 1977 AbstractA factorial set for the Gaussian integers is a set G = {g1, g2 … gn} of Gaussian integers such that G(z) = Πk (z − gk)gk takes Gaussian integer values at Gaussian integers. We characterize factorial sets and give a lower bound for max∥z∥2=nπ ∥ G(z)∥. It is conjectured that there are infinitely many factorial sets.
Douglas Hensley, Hensley, Douglas
openaire +3 more sources
A Fast Euclidean Algorithm for Gaussian Integers
Journal of Symbolic Computation, 2002 Euclid's algorithm with Gaussian integers is carried out using approximate division (leading parts only). Often the quotient has small components, and then addition and shifting can be used instead of multiplication, thereby speeding up the process. Experiments with randomly chosen numbers are reported, with tables showing the improvement obtained.
Collins, George E.
openaire +2 more sources
Gaussian Twin Neighborhood Prime Labeling on Fan Digraphs
Ratio Mathematica, 2023 Gaussian integers are complex numbers of the form \gamma=x+iy where x and y are integers and i^2=-1. The set of Gaussian integers is usually denoted by \mathbb{Z}[i].
K Palani, A Shunmugapriya
doaj +1 more source
Encoder Hurwitz Integers: Hurwitz Integers that have the “Division with Small Remainder” Property
Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 2023 Considering error-correcting codes over Hurwitz integers, prime Hurwitz integers are considered. On the other hand, considering transmission over Gaussian channel, Hurwitz integers, whose the norm is either a prime integer or not a prime integer, are ...
Ramazan Duran
doaj +1 more source
Fundamental Results of Cyclic Codes over Octonion Integers and Their Decoding Algorithm
Computation, 2022 Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection, error correction, data transmission, and data storage.
Muhammad Sajjad +4 more
doaj +1 more source
Generalized Concatenated Codes over Gaussian and Eisenstein Integers for Code-Based Cryptography
Cryptography, 2021 The code-based McEliece and Niederreiter cryptosystems are promising candidates for post-quantum public-key encryption. Recently, q-ary concatenated codes over Gaussian integers were proposed for the McEliece cryptosystem, together with the one-Mannheim ...
Johann-Philipp Thiers +1 more
doaj +1 more source