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Complex number whose real and imaginary parts are both integers
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition
Gaussian_integer
Mathematical table
A Gaussian integer is either the zero, one of the four units (±1, ±i), a Gaussian prime or composite. The article is a table of Gaussian Integers x +
Table of Gaussian integer factorizations
Table_of_Gaussian_integer_factorizations
Complex number whose mapping on a coordinate plane produces a triangular lattice
root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice
Eisenstein_integer
Integer side lengths of a right triangle
of a prime Gaussian integer if the hypotenuse is prime. If the Gaussian integer is not prime then it is the product of two Gaussian integers p and q with
Pythagorean_triple
Root of a quadratic polynomial with a unit leading coefficient
rational integers, such as 2 {\textstyle {\sqrt {2}}} , and the complex number i = − 1 {\textstyle i={\sqrt {-1}}} , which generates the Gaussian integers. Another
Quadratic_integer
Algorithm for computing greatest common divisors
generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. This led to modern abstract algebraic
Euclidean_algorithm
Condition under which an odd prime is a sum of two squares
of a Gaussian integer is an integer equal to the square of the absolute value of the Gaussian integer. The norm of a product of Gaussian integers is the
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Principal square root of minus 1
Gaussian integers. The sum, difference, or product of Gaussian integers is also a Gaussian integer: ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i
Imaginary_unit
Used to count, measure, and label
and imaginary parts of a complex number are both integers, then the number is called a Gaussian integer. The symbol for the complex numbers is C or C {\displaystyle
Number
Natural number
highly composite number, and the first colossally abundant number. An integer is determined to be even if it is divisible by two. When written in base
2
Algebraic construction
are often called the "rational integers" because of this. The next simplest example is the ring of Gaussian integers Z [ i ] {\displaystyle \mathbb {Z}
Ring_of_integers
Conditions in number theory
second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas. He said that a third monograph
Quartic_reciprocity
Type of complex number
qualified as quadratic integers. Gaussian integers, complex numbers a + bi for which both a and b are integers, are also quadratic integers. This is because
Algebraic_number
Mathematical functions
functions, sl and cl have a square period lattice (a multiple of the Gaussian integers) with fundamental periods { ( 1 + i ) ϖ , ( 1 − i ) ϖ } , {\displaystyle
Lemniscate_elliptic_functions
Probability distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued
Normal_distribution
Complex number that solves a monic polynomial with integer coefficients
number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic
Algebraic_integer
Aspect of algebraic number theory
= Q and L = Q(i), so OK is simply Z, and OL = Z[i] is the ring of Gaussian integers. Although this case is far from representative — after all, Z[i] has
Splitting of prime ideals in Galois extensions
Splitting_of_prime_ideals_in_Galois_extensions
Monochrome light beam whose amplitude envelope is a Gaussian function
optics, a Gaussian beam is an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function;
Gaussian_beam
Gives conditions for the solvability of quadratic equations modulo prime numbers
without using quartic reciprocity. For an odd Gaussian prime π {\displaystyle \pi } and a Gaussian integer α {\displaystyle \alpha } relatively prime to
Quadratic_reciprocity
Algorithm for solving systems of linear equations
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of
Gaussian_elimination
Commutative ring with a Euclidean division
integers. Define f (n) = |n|, the absolute value of n. Z[ i ], the ring of Gaussian integers. Define f (a + bi) = a2 + b2, the norm of the Gaussian integer
Euclidean_domain
Theory of a class of elliptic curves
such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice. It has an aspect belonging to the theory of
Complex_multiplication
Mathematical problem in number theory
complex plane, is it possible to "walk to infinity" in the Gaussian integers using the Gaussian primes as stepping stones and taking bounded-length steps
Gaussian_moat
algorithm Gaussian brackets – described on WolframMathWorld Gaussian's modular arithmetic Gaussian integer, usually written as Z[i] Gaussian prime Gaussian logarithms
List of things named after Carl Friedrich Gauss
List_of_things_named_after_Carl_Friedrich_Gauss
Set of integers, the lengths of the sides of a triangle with a 60° angle
such triangles to the Eisenstein integers is analogous to the relation of Pythagorean triples to the Gaussian integers. Triangles with an angle of 60°
Eisenstein_triple
Prime number of the form 2^n – 1
of "integers" on complex numbers instead of real numbers, like Gaussian integers and Eisenstein integers. If we regard the ring of Gaussian integers, we
Mersenne_prime
Product of two distinct primes ≡ 3 (mod 4)
3, for some integer t. Integers of this form are referred to as Blum primes. This means that the factors of a Blum integer are Gaussian primes with no
Blum_integer
Replacing a number with a simpler value
reporting many computations – especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as
Rounding
Mathematical function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f ( x ) = exp ( − x 2 ) {\displaystyle f(x)=\exp(-x^{2})}
Gaussian_function
Statistical model
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that
Gaussian_process
Characterization by prime factors of sums of two squares
form the integer sequence 0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, ... They form the set of all norms of Gaussian integers; their square
Sum_of_two_squares_theorem
Negative integer two units from the origin in mathematics
Gaussian integer, negative two can be factored as i × ( 1 + i ) 2 {\displaystyle i\times (1+i)^{2}} , where 1 + i {\displaystyle 1+i} is a Gaussian prime
−2
Integers have unique prime factorizations
introduced what is now called the ring of Gaussian integers, the set of all complex numbers a + bi where a and b are integers. It is now denoted by Z [ i ] . {\displaystyle
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Complex number with rational components
the field of Gaussian rationals is neither ordered nor complete (as a metric space). The Gaussian integers Z[i] form the ring of integers of Q(i). The
Gaussian_rational
Family of polynomials
In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs
Gaussian_binomial_coefficient
Generalization of algebraic integers
Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers
Hurwitz_quaternion
Integral of the Gaussian function, equal to sqrt(π)
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f ( x ) = e − x 2 {\displaystyle f(x)=e^{-x^{2}}}
Gaussian_integral
Random matrix with gaussian entries
side-length of a matrix. Always a positive integer. W N {\displaystyle W_{N}} : a matrix sampled from a Gaussian ensemble with size N × N {\displaystyle
Gaussian_ensemble
Describes statistically the splitting of primes in a given Galois extension of Q
of the prime ( 1 + i ) {\displaystyle (1+i)} and the invertible Gaussian integer − i {\displaystyle -i} ; we say that 2 "ramifies". For instance
Chebotarev_density_theorem
Number divisible only by 1 and itself
integers. Its prime elements are known as Gaussian primes. Not every number that is prime among the integers remains prime in the Gaussian integers;
Prime_number
Computation modulo a fixed integer
mathematics, modular arithmetic is a system of arithmetic operations for integers, differing from the usual ones in that numbers "wrap around" when reaching
Modular_arithmetic
Field (mathematics) generated by the square root of an integer
field of Gaussian rationals and the discriminant is − 4 {\displaystyle -4} . The reason for such a distinction is that the ring of integers of K {\displaystyle
Quadratic_field
Branch of number theory
}a_{n}(1/t)^{n}} The integers have only two units, 1 and −1. Other rings of integers may admit more units. The Gaussian integers have four units, the
Algebraic_number_theory
German polymath and scholar (1777–1855)
on biquadratic residues (1828, 1832) Gauss introduced the ring of Gaussian integers Z [ i ] {\displaystyle \mathbb {Z} [i]} , showed that it is a unique
Carl_Friedrich_Gauss
Natural number
smallest prime that is not a Chen prime. 43 is also a Wagstaff prime, a Gaussian prime, and a Heegner number. 43 is the fourth term of Sylvester's sequence
43_(number)
Four-dimensional number system
theorem in number theory, which states that every nonnegative integer is the sum of four integer squares. As well as being an elegant theorem in its own right
Quaternion
Shape with four equal sides and angles
arithmetic as addition with c {\displaystyle c} . The Gaussian integers, complex numbers with integer real and imaginary parts, form a square lattice in
Square
Natural number
is the 39th prime number, an emirp, an isolated prime, a Chen prime, a Gaussian prime, a safe prime, and an Eisenstein prime with no imaginary part and
167_(number)
Natural number
prime. A Fortunate prime. A circular prime. A prime number that is also a Gaussian prime (since it is of the form 4n + 3). A happy prime. A Higgs prime. A
79_(number)
more commonly used to denote the greatest integer less than or equal to x {\displaystyle x} . The Gaussian brackets notation is defined as follows: [
Gaussian_brackets
Largest integer that divides given integers
of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest
Greatest_common_divisor
Electromagnetic effect in physics
The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems
Quantum_Hall_effect
Assignment of an initial value for variable
the constructor parameters: Example: class GaussianInteger { private: int re; int im; public: GaussianInteger(int re = 0, int im = 0): re{re}, im{im} {}
Initialization_(programming)
Function used in signal processing
10^{-3}\\\hline \end{array}}} The Fourier transform of a Gaussian is also a Gaussian. Since the support of a Gaussian function extends to infinity, it must either
Window_function
(Mathematical) decomposition into a product
P; this is a matrix formulation of Gaussian elimination. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique (up to the
Factorization
Number with an integer power equal to 1
they are Gaussian integers (D = −1): see Imaginary unit. For four other values of n, the primitive roots of unity are not quadratic integers, but the
Root_of_unity
Quotient of two integers
integers, a numerator p and a nonzero denominator q. For example, 3 7 {\displaystyle {\tfrac {3}{7}}} is a rational number, as is every integer (for
Rational_number
Decomposition of an integer as a sum of positive integers
partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only
Integer_partition
Division with remainder of integers
include fields, polynomial rings in one variable over a field, and the Gaussian integers. The Euclidean division of polynomials has been the object of specific
Euclidean_division
Provides an asymptotic formula for counting the number of prime ideals of a number field
seen already for the Gaussian integers. There for any prime number p of the form 4n + 1, p factors as a product of two Gaussian primes of norm p. Primes
Landau_prime_ideal_theorem
Sufficient condition for polynomial irreducibility
irreducible. Here "whole real numbers" are ordinary integers and "whole complex numbers" are Gaussian integers; one should similarly interpret "real and complex
Eisenstein's_criterion
Result on density of prime numbers
over the Gaussian integers is an extension of the idea of the distribution of primes, but in this case on the complex plane. Thus, as Gaussian primes extend
Bertrand's_postulate
Number with a real and an imaginary part
geometric problem. Another example is the Gaussian integers; that is, numbers of the form x + iy, where x and y are integers, which can be used to classify sums
Complex_number
Abstract algebra concept
{\displaystyle R:=\{a+b\mathrm {i} \mid a,b\in \mathbb {Z} \}} be the ring of Gaussian integers. Then Frac ( R ) = { c + d i ∣ c , d ∈ Q } {\displaystyle \operatorname
Field_of_fractions
Natural number
composite member of the 19-aliquot tree with 65 a Blum integer since both 7 and 11 are Gaussian primes. the sum of three consecutive squares, 42 + 52 +
77_(number)
Family of solutions to related differential equations
are when α {\displaystyle \alpha } is an integer or a half-integer. When α {\displaystyle \alpha } is an integer, the resulting Bessel functions are often
Bessel_function
Analogue of a prime number in a commutative ring
rings: The integers ±2, ±3, ±5, ±7, ±11, ... in the ring of integers Z the complex numbers (1 + i), 19, and (2 + 3i) in the ring of Gaussian integers Z[i] the
Prime_element
Generalization of algebraic variety
[x]/(x^{2}{+}1))=\mathop {\rm {Spec}} (\mathbb {Z} [i]),} the spectrum of the Gaussian integers Z [ i ] {\displaystyle \mathbb {Z} [i]} , and the quadratic extension
Scheme_(mathematics)
Generalization of Gaussian distribution
The q-Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. It is one example of a
Q-Gaussian_distribution
{-7}})} . This ring is a unique factorization domain. Eisenstein integer Gaussian integer Conway, John Horton; Smith, Derek A. (2003), On Quaternions and
Kleinian_integer
Natural number
summatory function Φ ( n ) {\displaystyle \Phi (n)} over the first 10 integers, and the smallest number n {\displaystyle n} with exactly 7 solutions for
32_(number)
Difference between logarithm and harmonic series
disk in the complex plane containing at least k {\displaystyle k} Gaussian integers. The following bounds have been established: 1.819776 < δ < 1.819833
Euler's_constant
Exploring properties of the integers with complex analysis
generalized his arithmetic progressions theorem from integers to the ring of Gaussian integers Z [ i ] {\displaystyle \mathbb {Z} [i]} . In two papers
Analytic_number_theory
Sigmoid shape special function
function at ∞ {\displaystyle \infty } is exactly 1 {\displaystyle 1} (see Gaussian integral). At the real axis, erf ( z ) {\displaystyle \operatorname {erf}
Error_function
Method of describing higher-order polyhedra
domains over the complex numbers: the Eisenstein integers for the triangular GC family, and the Gaussian integers for the quadrilateral GC family. Operators
Conway_polyhedron_notation
Extension of the factorial function
definition of the gamma function, resulting in a Gaussian integral. In general, for non-negative integer values of n {\displaystyle n} we have: Γ ( 1 2
Gamma_function
Periodic set of points
abelian group of rank 2 n {\displaystyle 2n} . For example, the Gaussian integers Z [ i ] = Z + i Z {\displaystyle \mathbb {Z} [i]=\mathbb {Z} +i\mathbb
Lattice_(group)
Algebraic structure
(x^{k})} , Z [ i ] {\displaystyle \mathbb {Z} [i]} : the ring of Gaussian integers, Z [ ω ] {\displaystyle \mathbb {Z} [\omega ]} (where ω {\displaystyle
Principal_ideal_domain
Type of probability distribution
distribution are dominated by (i.e. decay at least as fast as) the tails of a Gaussian. This property gives subgaussian distributions their name. Often in analysis
Sub-Gaussian_distribution
Complex-valued arithmetic function
whose Dirichlet characters are all Gaussian integers (the Dirichlet characters of the number n are all Gaussian integers if and only if n is divisor of 240)
Dirichlet_character
Type of integral domain
UFDs. In particular, the integers (also see Fundamental theorem of arithmetic), the Gaussian integers and the Eisenstein integers are UFDs. If R is a UFD
Unique_factorization_domain
24-dimensional repeating pattern of points
constructions as complex lattices, over either the Eisenstein or Gaussian integers. The Leech lattice can also be constructed using the ring of icosians
Leech_lattice
Pair of integers related by their divisors
(66928, 66992) are two amicable pairs (sequence A359334 in OEIS). Gaussian integer amicable pairs exist, e.g. s(8008 + 3960i) = 4232 − 8280i and s(4232
Amicable_numbers
Algorithm for computing the greatest common divisor
for fast integer multiplication. The binary GCD algorithm has also been extended to domains other than natural numbers, such as Gaussian integers, Eisenstein
Binary_GCD_algorithm
Unsolved problem in mathematics
squares of squares" (PDF). Acta Arithmetica. Cain, Onno (2019). "Gaussian Integers, Rings, Finite Fields, and the Magic Square of Squares". arXiv:1908
Magic_square_of_squares
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
{\displaystyle n} is any positive integer and G {\displaystyle \operatorname {\mathbb {G} } } is the set of all Gaussian integers of the form ( − 1 ) a ± b −
Lemniscate_constant
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016. Chamberland, Marc. "Binary BBP-Formulae for Logarithms and Generalized Gaussian-Mersenne
23_(number)
Conditions under which the congruence x^3 equals p (mod q) is solvable
of higher powers" are the rings of integers of the cyclotomic number fields; the Gaussian and Eisenstein integers are the simplest examples of these.
Cubic_reciprocity
Unsolved problem about sums of powers
Prouhet-Tarry-Escott solutions over the Gaussian integers (though solutions to the Alpers-Tijdeman problem do not exhaust the Gaussian integer solutions to Prouhet-Tarry-Escott)
Prouhet–Tarry–Escott_problem
Subset of a ring that forms a ring itself
to a subring, denoted R[a1, a2, ..., an]. For example, the ring of Gaussian integers Z [ i ] {\displaystyle \mathbb {Z} [i]} is a subring of C {\displaystyle
Subring
In number theory, measure of non-unique factorization
{\displaystyle \mathbb {Z} [\omega ]} , respectively the integers, Gaussian integers, and Eisenstein integers, are all principal ideal domains (and in fact are
Ideal_class_group
Algorithm for determinants of integers
Bareiss, Erwin H. (1968), "Sylvester's Identity and multistep integer-preserving Gaussian elimination" (PDF), Mathematics of Computation, 22 (103): 565–578
Bareiss_algorithm
2-dimensional integer lattice
listed in the table below. Centered square number Euclid's orchard Gaussian integer Hexagonal lattice Quincunx Square tiling Conway, John; Sloane, Neil
Square_lattice
Application of geometry in number theory
numbers. The ring of integers in a number field can be embedded as a lattice in a higher dimensional space. The Gaussian integers, which are all a + i
Geometry_of_numbers
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in
Gaussian_period
Construction in group theory
as matrices with determinant −1 and integer coefficients, or as matrices with determinant 1 and Gaussian integer coefficients. This maps to the symmetries
Projective_linear_group
Natural number
the Prime 13 in the 13-Aliquot tree. a Blum integer, since its two prime factors, 3 and 31 are both Gaussian primes. a repdigit in base 5 (3335), and 30
93_(number)
Natural number
namely 3 and 47. Since those prime factors are Gaussian primes, this means that 141 is a Blum integer. a Hilbert prime Sometimes used as an acronym [1
141_(number)
Remarks: The double cover acts on a 28-dimensional lattice over the Gaussian integers. Order: 213 ⋅ 37 ⋅ 52 ⋅ 7 ⋅ 11 ⋅ 13 = 448345497600 Schur multiplier:
List_of_finite_simple_groups
exponentially modified Gaussian distribution, a convolution of a normal distribution with an exponential distribution, and the Gaussian minus exponential distribution
List of probability distributions
List_of_probability_distributions
GAUSSIAN INTEGER
GAUSSIAN INTEGER
Male
Russian
Variant spelling of Russian Afanasiy, AFANASY means "immortal."
Female
Russian
(Людмила) Russian feminine form of Czech/Russian Ludmil, LUDMILA means "people's favor."Â
Male
Russian
Variant spelling of Russian Irinei, IRINEY means "peaceful."
Male
Russian
Variant spelling of Russian Afanasiy, AFANASII means "immortal."
Male
Russian
Variant spelling of Russian Vasiliy, VASSILY means "king."
Male
Russian
Variant spelling of Russian Vikentiy, VIKENTI means "conquering."
Male
Russian
(Паша) Russian pet form of Czech/Russian Pavel, PASHA means "small."
Male
Russian
Variant spelling of Russian Vasiliy, VASILI means "king."
Boy/Male
Australian, French, German, Irish
Curly-headed
Female
Russian
(Russian Ева): Armenian and Russian form of Greek Eva, YEVA means "life."Â
Male
Russian
(Russian ИÑидор): Russian form of Greek Isidoros, ISIDOR means "gift of Isis."
Male
Russian
Variant spelling of Russian Gennadiy, GENNADY means "noble."
Male
Russian
Variant spelling of Russian Faddei, FADEI means "courageous."
Male
Russian
Variant spelling of Russian Aleksey, ALEXEY means "defender."
Male
Russian
Variant spelling of Russian Gennadiy, GENNADI means "noble."
Male
Russian
(РоÑÑ) Russian pet form of Czech/Russian Rostislav, ROSTYA means "usurp-glory."
Male
Russian
Variant spelling of Russian Vasiliy, VASILY means "king."
Male
Russian
Variant spelling of Russian Afanasiy, AFANASEI means "immortal."
Male
Russian
Variant spelling of Russian Arseniy, ARSENI means "virile."
Male
Russian
Variant spelling of Russian Arseniy, ARSENIY means "virile."
GAUSSIAN INTEGER
GAUSSIAN INTEGER
Surname or Lastname
English
English : variant spelling of Fawcett.French : diminutive of Fosse.
Boy/Male
Australian, French, Greek
Broad; Broad Shouldered
Boy/Male
Tamil
Sanmukha | ஸநà¯à®®à¯à®•à®¾
Boy/Male
Welsh
Son of Roderick.
Surname or Lastname
English or Scottish
English or Scottish : probably a variant of Witham or Whitton.
Boy/Male
Indian, Punjabi, Sikh
Friendly in the Proximity of God
Boy/Male
Indian, Punjabi, Sikh
Heart of God
Boy/Male
Hindu
To practice
Female
English
English variant spelling of French Denise, DENICE means "follower of Dionysos."
Boy/Male
Indian, Kannada, Sanskrit
Lord Shiva
GAUSSIAN INTEGER
GAUSSIAN INTEGER
GAUSSIAN INTEGER
GAUSSIAN INTEGER
GAUSSIAN INTEGER
n.
Prussian leather.
a.
Of or pertaining to Russia, its inhabitants, or language.
a.
Of or pertaining to Prussia.
n. sing. & pl.
A Russian, or the Russians.
n.
A Russian drink distilled from rye.
n.
A native or inhabitant of Prussia.
n.
A Russian river craft used for transporting freight.
a.
Of or pertaining to Lithuania (formerly a principality united with Poland, but now Russian and Prussian territory).
n.
One who, not being a Russian, favors Russian policy and aggrandizement.
n.
A kind of carp (Cyprinus gibelio); -- called also Prussian carp.
n.
A Russian weight, equal to forty Russian pounds or about thirty-six English pounds avoirdupois.
n.
A Russian copper coin. See Kopeck.
n.
A native or inhabitant of Russia; the language of Russia.
n.
Morbid dread of Russia or of Russian influence.
n.
The Russian variety of bagatelle.
n.
A Russian measure of length containing 3,500 English feet.
a.
Prussian; -- applied to certain astronomical tables published in the sixteenth century, founded on the principles of Copernicus, a Prussian.
n.
A Russian measure of length = 2 ft. 4.246 inches.
n.
A Russian village community.