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Gaussian Integers

Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, Yasunari Shidama
2013 Formalized Mathematics
Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers .  ...  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) .  ...

Polynomials which take Gaussian integer values at Gaussian integers

Douglas Hensley
1977 Journal of Number Theory
A factorial set (FS) for the 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.  ...

Algorithms for Gaussian integer arithmetic

B. F. Caviness, G. E. Collins
1976 Proceedings of the third ACM symposium on Symbolic and algebraic computation - SYMSAC '76
In this paper new algorithms are given for 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).  ...

Montgomery Reduction for Gaussian Integers

Malek Safieh, Jürgen Freudenberger
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.  ...

On sums over Gaussian integers

D. G. Hazlewood
1975 Transactions of the American Mathematical Society
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  ...  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

D. G. Hazlewood
1975 Transactions of the American Mathematical Society
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  ...  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]

Andrew V. Lelechenko
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.  ...

On automatic subsets of the Gaussian integers [article]

Wieb Bosma, Robbert Fokkink, Thijmen Krebs
2016 arXiv   pre-print
Suppose that $a$ and $b$ are multiplicatively independent 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]

Timothy Caley
2011 arXiv   pre-print
In 2007, Alpers and Tijdeman gave examples of solutions to the PTE problem over the 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]

Felix Sidokhine
2014 arXiv   pre-print
In this article we show that the form $x^2 + iy^2 + z^2 + iw^2$ represents all 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  .  ...

On Mignotte Secret Sharing Schemes over Gaussian Integers [article]

Diego Munuera-Merayo
2021 arXiv   pre-print
Mignotte, based on the Chinese remainder theorem over the ring of 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.  ...

Cyclotomic matrices over the Eisenstein and Gaussian integers

Gary Greaves
2012 Journal of Algebra
We classify all cyclotomic matrices over the Eisenstein and 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  ...

Generating normal numbers over Gaussian integers

2008 Acta Arithmetica
coefficients, and by z i we denote a numbering of the 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.  ...

Factor rings of the Gaussian integers

Cody Patterson, Kirby C. Smith, Leon Van Wyk
2004 South African Journal of Science and Technology
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  ...  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.  ...

Abundance of Matrices In Gaussian Integers [article]

Aninda Chakraborty
2020 arXiv   pre-print
In this paper we proved it for the ring of 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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