The divisibility of Gaussian integers by large Gaussian primes

1965 ◽  
Vol 32 (3) ◽  
pp. 503-509 ◽  
Author(s):  
J. H. Jordan
Keyword(s):  
Gaussian Integers ◽  
Gaussian Primes

Quasi-uniqueness of the set of "Gaussian prime plus one's"

2014 ◽  
Vol 10 (07) ◽  
pp. 1783-1790
Author(s):  
Jay Mehta ◽  
G. K. Viswanadham
Keyword(s):  
Acta Arith ◽  
Gaussian Integers ◽  
Arithmetical Functions ◽  
Set Of Uniqueness ◽  
Positive Integers ◽  
Complex Valued ◽  
Gaussian Primes

We recall the well-known notion of the set of uniqueness for arithmetical functions, introduced by Kátai and several other mathematicians like Indlekofer, Elliot and Hoffman, independently. We define its analogue for completely additive complex-valued functions over the set of non-zero Gaussian integers with some examples. We show that the set of "Gaussian prime plus one's" along with finitely many Gaussian primes of norm up to some constant K is a set of uniqueness with respect to Gaussian integers. This is analogous to Kátai's result in the case of positive integers [I. Kátai, On sets characterizing number theoretical functions, II, Acta Arith.16 (1968) 1–14].

scholarly journals A cubic analogue of the Friedlander–Iwaniec spin over primes

2021 ◽  
Author(s):  
Jori Merikoski
Keyword(s):  
Unit Group ◽  
Prime Ideals ◽  
Jacobi Symbol ◽  
Gaussian Integers ◽  
Ring Of Integers ◽  
New Feature ◽  
Definition Of ◽  
Gaussian Primes

AbstractIn 1998 Friedlander and Iwaniec proved that there are infinitely many primes of the form $$a^2+b^4$$ a 2 + b 4 . To show this they used the Jacobi symbol to define the spin of Gaussian integers, and one of the key ingredients in the proof was to show that the spin becomes equidistributed along Gaussian primes. To generalize this we define the cubic spin of ideals of $${\mathbb {Z}}[\zeta _{12}]={\mathbb {Z}}[\zeta _3,i]$$ Z [ ζ 12 ] = Z [ ζ 3 , i ] by using the cubic residue character on the Eisenstein integers $${\mathbb {Z}}[\zeta _3]$$ Z [ ζ 3 ] . Our main theorem says that the cubic spin is equidistributed along prime ideals of $${\mathbb {Z}}[\zeta _{12}]$$ Z [ ζ 12 ] . The proof of this follows closely along the lines of Friedlander and Iwaniec. The main new feature in our case is the infinite unit group, which means that we need to show that the definition of the cubic spin on the ring of integers lifts to a well-defined function on the ideals. We also explain how the cubic spin arises if we consider primes of the form $$a^2+b^6$$ a 2 + b 6 on the Eisenstein integers.

Voronoi summation formula for Gaussian integers

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

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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