Gaussian integers and sums of two squares

2019 ◽  
pp. 17-53
Keyword(s):  
Gaussian Integers ◽  
Sums Of Two Squares

Voronoi summation formula for Gaussian integers

2021 ◽  
Author(s):  
Debika Banerjee ◽  
Ehud Moshe Baruch ◽  
Daniel Bump
Keyword(s):  
Summation Formula ◽  
Gaussian Integers ◽  
Voronoi Summation

Sums of two squares in analytic rings

1999 ◽  
Vol 230 (2) ◽  
pp. 317-328 ◽  
Author(s):  
Jesús M. Ruiz
Keyword(s):  
Sums Of Two Squares

2733. On Sums of Two Squares

1957 ◽  
Vol 41 (338) ◽  
pp. 288
Author(s):  
A. Sutcliffe
Keyword(s):  
Sums Of Two Squares

On the set of divisors of Gaussian integers

2019 ◽  
Vol 50 (2) ◽  
pp. 355-366
Author(s):  
Helmut Maier ◽  
Saurabh Kumar Singh
Keyword(s):  
Gaussian Integers

scholarly journals Gaussian Integers

2013 ◽  
Vol 21 (2) ◽  
pp. 115-125
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Daichi Mizushima ◽  
Yasunari Shidama
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Rational Numbers ◽  
Gaussian Integer ◽  
Algebraic Integers ◽  
Quotient Field ◽  
Gaussian Integers ◽  
Integer Ring

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. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

scholarly journals Generator of PRN's on the norm group

2021 ◽  
Vol 25 (2(36)) ◽  
pp. 26-39
Author(s):  
P. Fugelo ◽  
S. Varbanets
Keyword(s):  
Prime Number ◽  
Quadratic Field ◽  
Exponential Sum ◽  
Random Numbers ◽  
Gaussian Integers ◽  
New Family ◽  
Serial Test ◽  
Polynomial Representations ◽  
Modulo P ◽  
Norm Group

Let $p$ be a prime number, $d\in\mathds{N}$, $\left(\frac{-d}{p}\right)=-1$, $m>2$, and let $E_m$ denotes the set of of residue classes modulo $p^m$ over the ring of Gaussian integers in imaginary quadratic field $\mathds{Q}(\sqrt{-d})$ with norms which are congruented with 1 modulo $p^m$. In present paper we establish the polynomial representations for real and imagimary parts of the powers of generating element $u+iv\sqrt{d}$ of the cyclic group $E_m$. These representations permit to deduce the ``rooted bounds'' for the exponential sum in Turan-Erd\"{o}s-Koksma inequality. The new family of the sequences of pseudo-random numbers that passes the serial test on pseudorandomness was being buit.

ELEMENTS OF ORDER FOUR IN THE NARROW CLASS GROUP OF REAL QUADRATIC FIELDS

2015 ◽  
Vol 100 (1) ◽  
pp. 21-32
Author(s):  
ELLIOT BENJAMIN ◽  
C. SNYDER
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Ideal Class Group ◽  
Quadratic Number ◽  
Gaussian Integers ◽  
Real Quadratic Fields ◽  
Order Four ◽  
Real Quadratic ◽  
Narrow Class

Using the elements of order four in the narrow ideal class group, we construct generators of the maximal elementary $2$-class group of real quadratic number fields with even discriminant which is a sum of two squares and with fundamental unit of positive norm. We then give a characterization of when two of these generators are equal in the narrow sense in terms of norms of Gaussian integers.

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