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Gaussian Integers

2013
*
Formalized Mathematics
*

*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. ...

*integers*, 1(∈ the set of

*Gaussian*

*integers*), 0(∈ the set of

*Gaussian*

*integers*) . ...

##
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Polynomials which take Gaussian integer values at Gaussian integers

1977
*
Journal of Number Theory
*

A factorial set (FS) for the

doi:10.1016/0022-314x(77)90012-9
fatcat:6qafj3a3hzbyvje7vyfxjr2ife
*Gaussian**integers*is a set of nonzero*Gaussian**integers*G = {g k : 1 & k < n} such that for each*Gaussian**integer*5, G(c) = l-I,"=, (5 -gk)/gk is a*Gaussian**integer*, i.e., ... Then for all*Gaussian**integers*5, G(5) = nzE1 (5 -gk)/& is a*Gaussian**integer*, so K(p, G(S)) > 0. ...##
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Algorithms for Gaussian integer arithmetic

1976
*
Proceedings of the third ACM symposium on Symbolic and algebraic computation - SYMSAC '76
*

In this paper new algorithms are given for

doi:10.1145/800205.806321
dblp:conf/issac/CavinessC76
fatcat:u3jzbgtjvjbsthpcyoyatrs5ma
*Gaussian**integer*division and the calculation of the greatest common divisor of two*Gaussian**integers*. ... Introduction A*Gaussian**integer*is a number of the form al+a2 i where a I and a 2 are (rational)*integers*and i = ~/-T. ... Now let b be another non-zero*Gaussian**integer*with n = L~(b). ...##
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Montgomery Reduction for Gaussian Integers

2021
*
Cryptography
*

*Gaussian*

*integers*are complex numbers where the real and imaginary parts are

*integers*. Rings over

*Gaussian*

*integers*are isomorphic to ordinary

*integer*rings. ... Typically, Montgomery reduction is used for rings of ordinary

*integers*. In contrast, we investigate the modular reduction over rings of

*Gaussian*

*integers*. ... arithmetic over

*Gaussian*

*integers*. ...

##
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On sums over Gaussian integers

1975
*
Transactions of the American Mathematical Society
*

As a consequence of general estimates, asymptotic estimates with explicit error terms for the number of

doi:10.1090/s0002-9947-1975-0371842-6
fatcat:kbq5nzi26fe25cssozdx6ucvie
*Gaussian**integers*with only "large" prime factors and for the number of*Gaussian**integers*with only ... The object of this paper is to give asymptotic estimates for some number theoretic sums over*Gaussian**integers*. ... Let G represent the set of*Gaussian**integers*and let P represent the set of*Gaussian*primes. ...##
###
On Sums over Gaussian Integers

1975
*
Transactions of the American Mathematical Society
*

As a consequence of general estimates, asymptotic estimates with explicit error terms for the number of

doi:10.2307/1997387
fatcat:nbp3toxxlba7vfzzmoiueqjzwq
*Gaussian**integers*with only "large" prime factors and for the number of*Gaussian**integers*with only ... The object of this paper is to give asymptotic estimates for some number theoretic sums over*Gaussian**integers*. ... Let G represent the set of*Gaussian**integers*and let P represent the set of*Gaussian*primes. ...##
###
Dirichlet divisor problem on Gaussian integers
[article]

2018
*
arXiv
*
pre-print

*integers*. ... estimates for the asymptotic behaviour of $\sum_{N \alpha \le x} \mathfrak{t}_k(\alpha)$, where $N$ stands for the norm of a complex number and $\mathfrak{t}_k$ is the $k$-dimensional divisor function on

*Gaussian*... It is natural to extend the notion of divisor functions from

*integers*to other unique factorisation domains such as rings of quadratic

*integers*. ...

##
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On automatic subsets of the Gaussian integers
[article]

2016
*
arXiv
*
pre-print

Suppose that $a$ and $b$ are multiplicatively independent

arXiv:1602.08579v3
fatcat:wd24r33gunfofc6cpvr4ldnwky
*Gaussian**integers*, that are both of modulus~$\geq \sqrt 5$. ... Numeration systems of the*Gaussian**integers*Not all*Gaussian**integers*b can be used to represent Z[i] as a (D, b) ring. If b is a unit then Rb is equal to R and there is only one residue class. ... For every*Gaussian**integer*|b| ≥ √ 5 there exists a D such that Z[i] is a (D, b)-ring. ...##
###
The Prouhet-Tarry-Escott problem for Gaussian integers
[article]

2011
*
arXiv
*
pre-print

In 2007, Alpers and Tijdeman gave examples of solutions to the PTE problem over the

arXiv:1011.1262v2
fatcat:hm6i3xuwobeovau4iphsksrksm
*Gaussian**integers*. This paper extends the framework of the problem to this setting. ... We prove generalizations of results from the literature, and use this information along with computational techniques to find ideal solutions to the PTE problem in the*Gaussian**integers*. ... Their article also gives an example of a solution to the pte problem over the*Gaussian**integers*, Z[i]. ...##
###
A Universal Quaternary Quadratic Form over Gaussian Integers
[article]

2014
*
arXiv
*
pre-print

In this article we show that the form $x^2 + iy^2 + z^2 + iw^2$ represents all

arXiv:1310.6293v2
fatcat:3hvtwhvp3vbalf4jgmic3ztwyq
*gaussian**integers*. ... The main tools used in this proof are Fermat's little theorem (over finite field extensions), the Mordell-Niven theorem (representation of some*gaussians*), and the generalized Euler-identity over finite ... By "universal form", we mean a form f with*gaussian**integer*coefficients in*gaussian**integer*variables which represent all*gaussian**integers*[6] . ...##
###
On Mignotte Secret Sharing Schemes over Gaussian Integers
[article]

2021
*
arXiv
*
pre-print

Mignotte, based on the Chinese remainder theorem over the ring of

arXiv:2104.06361v2
fatcat:m3os6ffup5dlhblktt4qmxqow4
*integers*. In this article we extend the Mignotte's scheme to the ring of*Gaussian**Integers*and study some of its properties. ... From now on we will focus on the extension to*Gaussian**Integers*. The ring of*Gaussian**Integers*. A*Gaussian**Integer*is a complex number z = a + bi, where both a and b are*integers*. ... Some background on SSS's and*Gaussian**Integers*First, we recall some known facts about secret sharing schemes and*Gaussian**Integers*. ...##
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Cyclotomic matrices over the Eisenstein and Gaussian integers

2012
*
Journal of Algebra
*

We classify all cyclotomic matrices over the Eisenstein and

doi:10.1016/j.jalgebra.2012.09.006
fatcat:blvzau5cjrdoveyyfrjvhapxt4
*Gaussian**integers*, that is, all Hermitian matrices over the Eisenstein and*Gaussian**integers*that have all their eigenvalues in the interval ... Classification of cyclotomic matrices over Z[ω] As with the classification over the*Gaussian**integers*, we split up the result to deal with uncharged graphs and charged graphs separately. Z[ω] . ... The Z-graph S 14 is also a Z[i]-graph. , S 7 , S 8 , and S 8 in Figs. 3, 4, and 8.The theorems above give a complete classification of cyclotomic matrices over the*Gaussian**integers*as follows.Theorem ...##
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Generating normal numbers over Gaussian integers

2008
*
Acta Arithmetica
*

coefficients, and by z i we denote a numbering of the

doi:10.4064/aa135-1-5
fatcat:zgn3bhls4zfrpnk4w6dgd5gt54
*Gaussian**integers*. ... A generalization of normal numbers to*Gaussian**integers*and their canonical number systems, which were characterized by Kátai and Szabó, is given. ... Let (b, D) be a CNS in the*Gaussian**integers*. ...##
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Factor rings of the Gaussian integers

2004
*
South African Journal of Science and Technology
*

Whereas the homomorphic images of Z (the ring of

doi:10.4102/satnt.v23i4.201
fatcat:mcm2hocfqbezhgpvimo63zcfmy
*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*... 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. ... SUMMARY In virtually every introductory abstract algebra text the ring Z[i] of*Gaussian**integers*is introduced. ...##
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Abundance of Matrices In Gaussian Integers
[article]

2020
*
arXiv
*
pre-print

In this paper we proved it for the ring of

arXiv:2010.10305v1
fatcat:guzuwakwkvgsdgzf2lxbndna5y
*Gaussian**integers*. We showed the result when the matrix is taken with entries from \mathbb{Q}\left[i\right]. ... We have extend these results of abunndance of matrices on Z [i], the ring of*Gaussian**integers*. In Z [i], there is no order relation. So, we need to derive this results using different technique. ...
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