Gaussian number

Gaussian numbers as grid points in the complex number plane

The Gaussian numbers (after Carl Friedrich Gauß ; English Gaussian integer ) are a generalization of the whole numbers in the complex numbers . Every Gaussian number lies on an integer coordinate point of the complex plane. The Gaussian numbers form the integral ring of the square number field . In addition, the Gaussian numbers form a Euclidean ring and thus in particular a factorial ring . ${\displaystyle \mathbb {Q} (\mathrm {i} )}$

A slightly more complicated generalization of integers that can also be embedded in the complex plane are the Eisenstein numbers .

Historical background

Gaussian numbers were given by Gauss in the treatise Theory of Biiquadratic Residues. Second treatise (1832, in Latin) first introduced.

The quadratic reciprocity law (which Gauss was able to prove for the first time in 1796) links the solvability of congruence with the solvability of . Likewise, the cubic reciprocity law links the solvability of congruence with that of, and the biquadratic reciprocity law is the link between with . ${\displaystyle x^{2}\equiv q{\pmod {p}}}$ ${\displaystyle x^{2}\equiv p{\pmod {q}}}$ ${\displaystyle x^{3}\equiv q{\pmod {p}}}$ ${\displaystyle x^{3}\equiv p{\pmod {q}}}$ ${\displaystyle x^{4}\equiv q{\pmod {p}}}$ ${\displaystyle x^{4}\equiv p{\pmod {q}}}$

Gauss found out that the biquadratic reciprocity law and the additions to it are much easier to formulate and prove than statements about “whole complex numbers” (i.e. Gaussian numbers).

In a footnote (p. 541) he mentions that the Eisenstein numbers are the natural range for theorems about cubic reciprocity and similar extensions of the integers are the suitable ranges for the investigation of higher powers.

This treatise contains not only the introduction of Gaussian numbers, but also the terms norm, unit, primary and associated, which are standard in algebraic number theory today.

definition

A Gaussian number is through ${\displaystyle g}$

${\displaystyle g=a+b{\mathrm {i} }}$

given, where and are integers. ${\displaystyle a}$ ${\displaystyle b}$

The ring of Gaussian numbers is also called the Gaussian number ring and is denoted by. He comes from by adjunction of the imaginary unit . ${\displaystyle \mathbb {Z} [\mathrm {i} ]}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathrm {i} }$

The Gaussian numbers are the points with integer coordinates in the Gaussian plane of numbers . They form a two-dimensional grid .

Prime elements

The spectrum of illustrates these relationships: The blobs correspond to the prime elements in the ring of Gaussian numbers that appear in the factorization of the prime number given below. ${\displaystyle \mathbb {Z} [\mathrm {i} ]}$

Prime elements in the complex plane. By multiplying with the units, the rotational symmetry is created by 90 °. Because there are also prime elements, the prime elements are also symmetrical to the bisector between the real and imaginary axes. ${\displaystyle a+\mathrm {i} b}$ ${\displaystyle \pm (b+\mathrm {i} a)}$ ${\displaystyle a=\pm b}$

As in any ring one can - analogously to  - also operate in number theory . In particular, prime elements can be defined as a generalization of the term prime number . The uniqueness of the prime factor representation then also applies to the Gaussian numbers. The prime elements in the ring of Gaussian numbers are, apart from the unit factors, exactly the prime numbers of the form , the element and the elements for which there is a prime number that can be written as . ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} [\mathrm {i} ]}$ ${\displaystyle \mathbb {Z} [\mathrm {i} ]}$ ${\displaystyle \pm 1,\pm \mathrm {i} }$ ${\displaystyle 4k+3,\ k\in \mathbb {N} _{0}}$ ${\displaystyle 1+i}$ ${\displaystyle a+b\cdot \mathrm {i} ,\ a,b\in \mathbb {Z} }$ ${\displaystyle a^{2}+b^{2}=p}$ ${\displaystyle p=4k+1,\ k\in \mathbb {Z} }$

The prime elements in the ring of Gaussian numbers are closely related to the ordinary prime numbers. They are divided into three classes (except for association, i.e. except for multiplication with and , the units of the ring of Gaussian numbers): ${\displaystyle \pm 1}$ ${\displaystyle \pm \mathrm {i} }$

The double prime factor of 2:

The number 2 can be written as the product of the prime elements and , which however only differ by one unit. So the prime factor decomposition applies - except for the fact that the factors are associated ${\displaystyle 1+\mathrm {i} }$ ${\displaystyle 1-\mathrm {i} }$ ${\displaystyle 1+\mathrm {i} =\mathrm {i} \cdot (1-\mathrm {i} )}$ ${\displaystyle (1+\mathrm {i} )(1-\mathrm {i} )=\mathrm {i} ^{3}\cdot (1+\mathrm {i} )^{2}}$

${\displaystyle 2=\mathrm {i} ^{3}\cdot (1+\mathrm {i} )^{2}}$

shows that 2 is associated to the square of the prime element (2 is branched ). ${\displaystyle 1+\mathrm {i} }$

Factors of prime numbers of the form 4 k + 1:

If a prime number is of the form with a natural number , it can be written in an essentially unambiguous way as the sum of two square numbers (see the two-squares theorem ): ${\displaystyle p}$ ${\displaystyle 4k+1}$ ${\displaystyle k}$ ${\displaystyle p}$

${\displaystyle p=a^{2}+b^{2}}$ with certain ${\displaystyle a,b\in \mathbb {Z} }$

Then

${\displaystyle p=(a+b\mathrm {i} )(a-b\mathrm {i} )}$

the prime factorization of , itself is therefore not a prime element in the ring of Gaussian numbers, but the product of two conjugated prime elements ( is decomposed ). For example is not a prime element, but and are two prime elements. ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle 5=(2+\mathrm {i} )(2-\mathrm {i} )}$ ${\displaystyle 2+\mathrm {i} }$ ${\displaystyle 2-\mathrm {i} }$

Prime numbers of the form 4 k + 3:

If a prime number is of the form with a natural number , then there is also a prime element in the ring of Gaussian numbers ( remains prime, it is inert ). ${\displaystyle p}$ ${\displaystyle 4k+3}$ ${\displaystyle k}$ ${\displaystyle p}$ ${\displaystyle p}$

The three cases describe the behavior of prime elements when the field of rational numbers is expanded to form the field of Gaussian numbers (created by the adjunct of the imaginary unit).

Prime factorization

A prime factorization for any Gaussian number that is unambiguous except for the order of the factors results z. For example, if one sets and selects the so-called primary clearly determined by the requirement (see below and congruences and residual classes ) from the four associated elements of each odd prime element and sorts them according to their norm: ${\displaystyle z\neq 0}$ ${\displaystyle p_{1}=1+\mathrm {i} }$ ${\displaystyle p_{m}}$ ${\displaystyle p_{m}\equiv 1{\pmod {2+2\mathrm {i} }}}$

${\displaystyle {\begin{aligned}&p_{2}=-1+2\mathrm {i} ,\quad p_{3}=-1-2\mathrm {i} ,\quad p_{4}=-3,\quad p_{5}=+3+2\mathrm {i} ,\quad \\&p_{6}=+3-2\mathrm {i} ,\quad p_{7}=+1+4\mathrm {i} ,\quad p_{8}=+1-4\mathrm {i} ,\;\dots ,\;p_{15}=-7,\dotsc \end{aligned}}}$

(obviously the natural prime numbers of the form are always to be provided with a negative sign, because ). The above definition obviously fulfills an important criterion: the product of any primary Gaussian number is also a primary number . So you get ${\displaystyle 4k+3}$ ${\displaystyle 4k+3\equiv 3\equiv -1{\pmod {2+2\mathrm {i} }}}$

${\displaystyle z={\mathrm {i} }^{k}\prod _{m\in \mathbb {N} }p_{m}^{\nu _{m}}}$with and (which of course only applies to a finite number of exponents ). ${\displaystyle k=0,\dotsc ,3}$ ${\displaystyle \nu _{m}\geq 0}$ ${\displaystyle \nu _{m}>0}$

Another, frequently used prime factor representation is obtained if the superfluous factors are omitted and only the prime divisors of are taken into account, i.e. H. all with . These are the numbers . This reads the representation ${\displaystyle p_{k}^{0}=1}$ ${\displaystyle z}$ ${\displaystyle p_{m}}$ ${\displaystyle \nu _{m}>0}$ ${\displaystyle q_{n}\in \{p_{1},\dots \},\;n=1,\dots ,r}$

${\displaystyle z={\mathrm {i} }^{k}\prod _{n=1}^{r}q_{n}^{\mu _{n}}}$with and ${\displaystyle k=0,\dotsc ,3}$ ${\displaystyle \mu _{n}>0}$

Euclidean Algorithm and Greatest Common Divisor (GCD)

Every Gaussian number has four associates , which are formed by multiplying with the units and which lie in all four quadrants of the complex number plane. A greatest common divisor (GCF) of two Gaussian numbers is defined as a Gaussian number with the following two properties: ${\displaystyle g\neq 0}$ ${\displaystyle \pm g,\pm \mathrm {i} g}$
${\displaystyle a,b}$ ${\displaystyle t}$

1. ${\displaystyle t|a}$and ie: is a common factor of and . ${\displaystyle t|b,\,}$ ${\displaystyle t}$ ${\displaystyle a}$ ${\displaystyle b}$
2. From and follows that is: Every common factor of and also divides . ${\displaystyle s|a}$ ${\displaystyle s|b}$ ${\displaystyle s|t,\,}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle t}$

It follows from this: All Gaussian numbers with these properties (given ) are associated. The gcd is thus essentially (except for associated) a clearly defined Gaussian number with the usual notation . ${\displaystyle t}$ ${\displaystyle a,b}$ ${\displaystyle t=(a,b)}$

If the prime factorization of and is known, that is, the GCF is of course immediately given by with . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a=i^{k}\prod _{m}p_{m}^{\nu _{m}},\,b=i^{n}\prod _{m}p_{m}^{\mu _{m}}}$ ${\displaystyle (a,b)=\prod _{m}p_{m}^{\lambda _{m}}}$ ${\displaystyle \lambda _{m}=\mathrm {min} (\nu _{m},\mu _{m})}$

Illustration of the Euclidean algorithm

Otherwise you can use the Euclidean algorithm : To determine the GCF of two numbers , it works similarly to that for whole numbers. It applies to everyone (in particular ). And there is a pair of Gaussian numbers for ${\displaystyle z_{0},z_{1}}$ ${\displaystyle (z_{0},0)=z_{0}}$ ${\displaystyle z_{0}}$ ${\displaystyle (0,0)=0}$ ${\displaystyle z_{1}\neq 0}$ ${\displaystyle q_{1},z_{2}}$

${\displaystyle z_{0}=q_{1}z_{1}+z_{2}}$ and ${\displaystyle |z_{2}|<|z_{1}|.}$

It is determined to be the Gaussian number that is closest to the fraction . The following always applies to and , so and consequently . ${\displaystyle q_{1}=m+n\mathrm {i} }$ ${\displaystyle \xi :={\frac {z_{0}}{z_{1}}}}$ ${\displaystyle \left|m-\operatorname {Re} (\xi )\right|\leq {\frac {1}{2}}}$ ${\displaystyle \left|n-\operatorname {Im} (\xi )\right|\leq {\frac {1}{2}}}$ ${\displaystyle d:=\left|q_{1}-\xi \right|\leq d_{max}={\frac {1}{\sqrt {2}}}}$ ${\displaystyle |z_{2}|\leq {\frac {|z_{1}|}{\sqrt {2}}}}$

If so, continue with and etc. until . Then the sought GCD: . ${\displaystyle z_{2}\neq 0}$ ${\displaystyle z_{1}=q_{2}z_{2}+z_{3}}$ ${\displaystyle |z_{3}|<|z_{2}|}$ ${\displaystyle z_{n+1}=0}$ ${\displaystyle z_{n}}$ ${\displaystyle (z_{0},z_{1})=z_{n}}$

Example:
Find the gcd of the Gaussian numbers . The quotient is . The four Gaussian numbers come into question for. We choose e.g. B. and received . The next step is , i. That is, the rest is : The algorithm breaks off and we get the GCD . ${\displaystyle z_{0}=5+\mathrm {i} ,\;z_{1}=2}$ ${\displaystyle {\frac {z_{0}}{z_{1}}}=2{,}5+0{,}5\mathrm {i} }$ ${\displaystyle q_{1}}$ ${\displaystyle 2,2+\mathrm {i} ,3,3+\mathrm {i} }$ ${\displaystyle q_{1}=2}$ ${\displaystyle z_{2}=z_{0}-q_{1}z_{1}=5+\mathrm {i} -4=1+\mathrm {i} }$ ${\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {2}{1+\mathrm {i} }}=1-\mathrm {i} }$ ${\displaystyle z_{3}=0}$ ${\displaystyle {\underline {(5+\mathrm {i} ,2)=z_{2}=1+\mathrm {i} }}}$

Congruences and remainder classes

Two Gaussian numbers are said to be congruent with respect to a Gaussian modulus if there is a Gaussian number with . You write for it . Then there is also a common rest with . As above, the factors can be determined in such a way that applies. ${\displaystyle z_{1},z_{2}}$ ${\displaystyle z_{0}}$ ${\displaystyle q}$ ${\displaystyle z_{1}-z_{2}=qz_{0}}$ ${\displaystyle z_{1}\equiv z_{2}{\pmod {z_{0}}}}$ ${\displaystyle r}$ ${\displaystyle z_{1}=q_{1}z_{0}+r,\,z_{2}=q_{2}z_{0}+r}$ ${\displaystyle q_{1},q_{2}}$ ${\displaystyle |r|\leq {\frac {|z_{0}|}{\sqrt {2}}}}$

The congruence ${\displaystyle R_{z_{0}}}$ to the modulus induced in the Gaussian ring a classification . One defines as the set of Gaussian numbers , for which: . The set is called a remainder class modulo . The following applies: ${\displaystyle z_{0}}$ ${\displaystyle \mathbb {Z} [\mathrm {i} ]}$ ${\displaystyle \mathbb {Z} [\mathrm {i} ]/R_{z_{0}}}$ ${\displaystyle {\bar {a}}}$ ${\displaystyle z}$ ${\displaystyle z\equiv a{\pmod {z_{0}}}}$ ${\displaystyle {\bar {a}}}$ ${\displaystyle z_{0}}$

${\displaystyle {\bar {a}}={\bar {b}}}$ exactly when ${\displaystyle a\equiv b{\pmod {z_{0}}}}$

Adding and multiplying congruences are very simple: From and it follows: ${\displaystyle a_{1}\equiv b_{1}{\pmod {z_{0}}}}$ ${\displaystyle a_{2}\equiv b_{2}{\pmod {z_{0}}}}$

${\displaystyle a_{1}+a_{2}\equiv b_{1}+b_{2}{\pmod {z_{0}}}}$

${\displaystyle a_{1}\cdot a_{2}\equiv b_{1}\cdot b_{2}{\pmod {z_{0}}}}$

That shows the definitions

${\displaystyle {\bar {a}}+{\bar {b}}:={\overline {a+b}}}$

${\displaystyle {\bar {a}}\cdot {\bar {b}}:={\overline {ab}}}$

for the sum and the product of remainder classes are well-defined (i.e. independent of the representative) and are therefore justified. With these operations, the set of remainder classes then forms a commutative ring with a zero element and a one element , the so-called remainder class ring modulo . ${\displaystyle \mathbb {Z} [\mathrm {i} ]/R_{z_{0}}}$ ${\displaystyle {\bar {0}}}$ ${\displaystyle {\bar {1}}}$ ${\displaystyle z_{0}}$

Examples:

1. There are exactly two residual classes for the module , namely the main ideal of all multiples of the module and , which form a checkerboard pattern in the Gaussian number plane. They can be viewed as an extension of the even or odd natural numbers and can therefore be referred to as (in) even Gaussian numbers (Gaussian divides the even numbers into semi-even and even, i.e. divisible by 2). ${\displaystyle z_{0}=1+\mathrm {i} }$ ${\displaystyle {\bar {0}}=\{0,\pm 2,\pm 4,\dotsc ,\pm 1\pm \mathrm {i} ,\pm 3\pm \mathrm {i} ,\dotsc \}}$ ${\displaystyle z\cdot (1+\mathrm {i} )}$ ${\displaystyle {\bar {1}}=\{\pm 1,\pm 3,\dotsc ,\pm \mathrm {i} ,\pm 2\pm \mathrm {i} ,\dotsc \}}$
   
2. There are exactly four remainder classes for the Gaussian module , namely . (Note that e.g. holds.) ${\displaystyle z_{0}=2}$ ${\displaystyle {\bar {0}},{\bar {1}},{\bar {\mathrm {i} }},{\overline {1+\mathrm {i} }}}$ ${\displaystyle 1+\mathrm {i} \equiv \pm 1\pm \mathrm {i} {\pmod {2}}}$

Complete residual systems

All 13 remainder classes with their minimal remnants (blue dots) in a square (marked in light green) to the module . A remainder class with is e.g. B. highlighted by orange / yellow dots. ${\displaystyle Q_{00}}$ ${\displaystyle z_{0}=3+2\mathrm {i} }$ ${\displaystyle z=2-4\mathrm {i} \equiv {-i}{\pmod {z_{0}}}}$

In order to determine all residual classes for a module , a square grid can be placed over the complex number plane with the illustration . The grid lines are the straight lines with and or . You divide the plane into squares (with integers ) . The four corner points of are the associated points . If there is an even Gaussian number, then all four are Gaussian numbers (and also congruent to each other), otherwise none. In the first case we take e.g. B. only the corner point as belonging. Within each square all Gaussian numbers are incongruent if one excludes the upper limits: (if Gaussian numbers are on the boundary lines, then always pairwise congruent numbers). ${\displaystyle z_{0}}$ ${\displaystyle z(s,t)=(s+\mathrm {i} t)z_{0}}$ ${\displaystyle s=\pm {\frac {1}{2}},\pm {\frac {3}{2}},\dotsc }$ ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle t=\pm {\frac {1}{2}},\pm {\frac {3}{2}},\dotsc ,s\in \mathbb {R} }$ ${\displaystyle Q_{mn}}$ ${\displaystyle m,n}$ ${\displaystyle s=\left[m-{\frac {1}{2}},m+{\frac {1}{2}}\right),t=\left[n-{\frac {1}{2}},n+{\frac {1}{2}}\right)}$ ${\displaystyle Q_{00}}$ ${\displaystyle {\frac {\pm 1\pm i}{2}}z_{0}}$ ${\displaystyle z_{0}}$ ${\displaystyle {\frac {-1-\mathrm {i} }{2}}z_{0}}$ ${\displaystyle Q_{00}}$ ${\displaystyle s<m+{\frac {1}{2}},t<n+{\frac {1}{2}}}$

The square thus describes all minimal remainders, in the sense that all other elements in the remainder classes are not smaller in terms of amount (Gauss calls them the absolutely smallest remainder ). ${\displaystyle Q_{00}}$

From this it can be deduced with simple geometric considerations that the number of remainder classes for a given module is equal to its norm (with natural numbers the number of remainder classes for a module is trivially equal to the amount ). ${\displaystyle z_{0}=m+n\mathrm {i} }$ ${\displaystyle N(z_{0}):=|z_{0}|^{2}=m^{2}+n^{2}}$ ${\displaystyle m}$ ${\displaystyle |m|}$

You can see immediately that all squares are congruent (including the grid points). They have the side length , i.e. the area, and all have the same number of Gaussian numbers, which we denote with . In general, the number of grid points in any square of the area is determined by . If we now consider a large square made of squares , there are always grid points in it. So what counts is what results in the Limes . ${\displaystyle |z_{0}|}$ ${\displaystyle F=|z_{0}|^{2}=m^{2}+n^{2}}$ ${\displaystyle N_{g}}$ ${\displaystyle A}$ ${\displaystyle A+O({\sqrt {A}})}$ ${\displaystyle k\times k}$ ${\displaystyle Q_{mn}}$ ${\displaystyle k^{2}F+O(k{\sqrt {F}})}$ ${\displaystyle k^{2}N_{g}=k^{2}F+O(k{\sqrt {F}})}$ ${\displaystyle k\to \infty }$ ${\displaystyle {\underline {N_{g}=F=m^{2}+n^{2}}}}$

Prime residual class group and Euler's Phi function

Many theorems (and proofs) for modules of integers can be transferred directly to modules of Gaussian numbers by replacing the modulus with the norm. This applies in particular to the prime remainder class group and the Fermat-Euler theorem, as will be briefly added here.

The prime residual class group (pRG) of the residual class ring modulo is the multiplicative group of its units. It consists of all residue classes with too divider foreign , ie which: . The number of its elements is denoted as (analogous to Euler's Phi function for whole numbers ). For prime elements it results immediately and for any (composite) Gaussian numbers one can use Euler's product formula ${\displaystyle z}$ ${\displaystyle {\bar {a}}}$ ${\displaystyle z}$ ${\displaystyle a}$ ${\displaystyle (a,z)=1}$ ${\displaystyle \phi (z)}$ ${\displaystyle \varphi (m)}$ ${\displaystyle m}$ ${\displaystyle \phi (p)=|p|^{2}-1}$ ${\displaystyle z}$

${\displaystyle \phi (z)=|z|^{2}\prod _{p_{m}|z}\left(1-{\frac {1}{|p_{m}|^{2}}}\right)}$

derive, where the product is to extend over all prime divisors of (with ). ${\displaystyle z=\mathrm {i} ^{k}\prod _{m}p_{m}^{\nu _{m}}}$ ${\displaystyle \nu _{m}>0}$

The important sentence from Fermat-Euler can also be transferred immediately:

From follows . ${\displaystyle (a,z)=1}$ ${\displaystyle a^{\phi (z)}\equiv 1{\pmod {z}}}$

With the help of this sentence you can z. B. solve some Diophantine equations for Gaussian numbers explicitly. For example, as solutions of the linear equation ${\displaystyle x,y}$

${\displaystyle ax+by=c}$

searched for given Gaussian numbers . For this you can o. B. d. A. Assume that every common divisor of and must also be a divisor of (otherwise the equation has no solution) and can therefore be eliminated. ${\displaystyle a,b,c}$ ${\displaystyle (a,b)=1}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$

To do this, consider this equation modulo what results . The Fermat-Euler theorem then provides an explicit solution , namely ${\displaystyle b}$ ${\displaystyle ax\equiv c{\pmod {b}}}$ ${\displaystyle x}$

${\displaystyle x\equiv ca^{\phi (b)-1}{\pmod {b}}}$,

d. H. all Gaussian numbers of the form with arbitrary Gaussian factors . Inserted into the output equation, this results ${\displaystyle x=ca^{\phi (b)-1}+ub}$ ${\displaystyle u}$

${\displaystyle y=c{\frac {1-a^{\phi (b)}}{b}}-ua}$,

which is also a Gaussian number according to the Fermat-Euler theorem.

Unsolved problems

The distribution of the Gaussian prime numbers in the plane

Most of the unsolved problems have to do with the distribution of Gaussian primes in the plane.

• The Gaussian circuit Problem (engl. Gauss's circle problem) is not concerned with Gaussian numbers per se , but asks for the number of lattice points within a circle with a given radius about the origin. This is equivalent to determining the number of Gaussian numbers with the norm less than a given value.

Two unsolved problems about Gaussian primes are e.g. B.

• On the real and imaginary coordinate lines there are an infinite number of Gaussian primes 3, 7, 11, 19 ... and their associated ones. Are there other straight lines on which there are an infinite number of prime numbers? In particular: Are there infinitely many prime numbers of the form ? ${\displaystyle 1+k\mathrm {i} }$

• Is it possible to wander through the level of Gaussian numbers to infinity by using the Gaussian prime numbers as support points and only taking steps of limited length? This is known as Gaussian grave problem (engl Gaussian moat problem.) Is known; it was drawn up in 1962 by Basil Gordon and is still unsolved.

literature

• Peter Bundschuh : Introduction to Number Theory . 6th, revised and updated edition. Springer-Verlag, Berlin a. a. 2008, ISBN 978-3-540-76490-8 .
• Harald Scheid : Number Theory . 3. Edition. Spectrum Academic Publishing House, Heidelberg u. a. 2003, ISBN 3-8274-1365-6 .

Web links

Commons : Gaussian number  - collection of images, videos, and audio files

Individual evidence

1. ↑ Harald Scheid: Number theory . 3. Edition. Spectrum Academic Publishing House, Heidelberg u. a. 2003, ISBN 3-8274-1365-6 , pp. 108 .
2. ↑ H. Maser (Ed.): Carl Friedrich Gauss' Arithmetic Studies on Higher Arithmetic. Springer, Berlin 1889, p. 534 ff.
3. ↑ Gaussian number . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
4. ↑ Peter Bundschuh: Introduction to Number Theory . 6th, revised and updated edition. Springer-Verlag, Berlin a. a. 2008, ISBN 978-3-540-76490-8 , pp. 76 .
5. ↑ Holger Brenner: Lecture . (PDF; 79 kB), Osnabrück University.
6. ↑ Herbert Pieper: The complex numbers . VEB Deutscher Verlag der Wissenschaften, Berlin 1988, ISBN 3-326-00406-0 , p. 119 .
7. ↑ E. Krätzel: Number theory . VEB Deutscher Verlag der Wissenschaften, Berlin 1981, p. 17 .
8. ↑ Ribenboim, Ch.III.4.D Ch. 6.II, Ch. 6.IV (Hardy & Littlewood's conjecture E and F)
9. ↑ Ellen Gethner, Stan Wagon, Brian Wick: A stroll through the Gaussian primes . In: American Mathematical Monthly . tape 105 , no. 4 , 1998, pp. 327-337 , doi : 10.2307 / 2589708 (English, Mathematical Reviews 1,614,871, zbMATH 0946.11002).
10. ^ Richard K. Guy: Unsolved problems in number theory . 3. Edition. Springer , 2004, ISBN 978-0-387-20860-2 , pp. 55-57 (English, zbMATH 1058.11001).