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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
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
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
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
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
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
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
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
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 polynomial
Algebraic_integer
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 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
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. The
Number
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
Generalization of Gaussian integers to quaternions
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
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
Family of polynomials
In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian numbers, Gaussian polynomials, or q-binomial coefficients)
Gaussian_binomial_coefficient
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
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
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
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
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
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
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
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
Sporadic simple group
125). Its double cover acts on a 28-dimensional lattice over the Gaussian integers. The lattice has 4×4060 minimal vectors; if minimal vectors are identified
Rudvalis_group
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
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
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
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 over the entire real line. Named after the German
Gaussian_integral
Number divisible only by 1 and itself
elements are known as Gaussian primes. Not every number that is prime among the integers remains prime in the Gaussian integers; for instance, the number
Prime_number
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
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
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
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
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
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
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
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
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
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
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
{-7}})} . This ring is a unique factorization domain. Eisenstein integer Gaussian integer Conway, John Horton; Smith, Derek A. (2003), On Quaternions and
Kleinian_integer
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
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
Assignment of an initial value for variable
constructor parameters: Example: class GaussianInteger { private: int re; int im; public: explicit GaussianInteger(int re = 0, int im = 0): re{re}, im{im}
Initialization (computer programming)
Initialization_(computer_programming)
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
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
(Mathematical) decomposition into a product
the integers called algebraic integers. The first ring of algebraic integers that have been considered were Gaussian integers and Eisenstein integers, which
Factorization
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)
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
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
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
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
Shape with four equal sides and angles
complex number arithmetic as addition with c. The Gaussian integers, complex numbers with integer real and imaginary parts, form a square lattice in
Square
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
Prime number congruent to 1 mod 4
squares involves Gaussian integers, the complex numbers whose real part and imaginary part are both integers. The norm of a Gaussian integer x + i y {\displaystyle
Pythagorean_prime
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
German polymath and scholar (1777–1855)
on biquadratic residues (1828, 1832) Gauss introduced the ring of Gaussian integers ℤ[i], showed that it is a unique factorization domain, and generalized
Carl_Friedrich_Gauss
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
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
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
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
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
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
Natural number
also a safe prime, a Woodall prime, a Thabit prime, a Fortunate prime, a Gaussian prime, an isolated prime, a long prime, a prime divisor of Mersenne numbers
383_(number)
Natural number
imaginary part and real part of the form 3 n − 1 {\displaystyle 3n-1} and a Gaussian prime with no imaginary part and real part of the form 4 n − 1 {\displaystyle
300_(number)
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
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
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
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
2026 release of AI-generated mathematical solutions
O(log(log(b))) terms. There are Gaussian moats of arbitrary width. An incomplete version of Artin's conjecture on primitive roots: every integer that is neither −1
OpenAI bulk release of mathematical papers
OpenAI_bulk_release_of_mathematical_papers
373, 379, 383, 397 (OEIS: A046066) Gaussian primes are primes p of the form p = 4k + 3, for a non-negative integer k. 3, 7, 11, 19, 23, 31, 43, 47, 59
List_of_prime_numbers
Type of mathematical expression
including the classical Gaussian elimination. A polynomial equation for which one is interested only in the solutions which are integers is called a Diophantine
Polynomial
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
Natural number
member of the 19-aliquot tree with 65 a Blum integer since its prime factorizations, 7 and 11, are both Gaussian primes. the sum of three consecutive squares
77_(number)
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
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)
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
Division with remainder of integers
– is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder
Euclidean_division
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
Inverse functions of sin, cos, tan, etc.
Abramowitz & Stegun 1972, p. 73, 4.3.44 Calcut, Jack S. (2009). "Gaussian integers and arctangent identities for pi". Am. Math. Monthly. 116 (6): 515–530
Inverse trigonometric functions
Inverse_trigonometric_functions
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
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
Mathematical function
computational chemistry and molecular physics, Gaussian orbitals (also known as Gaussian type orbitals, GTOs or Gaussians) are functions used as atomic orbitals
Gaussian_orbital
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
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
Family of probability distributions
=[n]_{q}[n-1]_{q}\cdots [2]_{q}\,} for an integer n > 2 and [1]q! = [0]q! = 1. The cumulative distribution function of the Gaussian q-distribution is given by G q
Gaussian_q-distribution
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
Generalization of algebraic variety
such as: V(x2+1)=Spec(ℤ[x]/(x2+1))=Spec(ℤ[i]), the spectrum of the Gaussian integers ℤ[i], and the quadratic extension ℤ⊂ℤ[i] corresponds to a branched
Scheme_(mathematics)
Formula for systems of linear equations
the Gaussian method. Therefore, if approximate calculation methods are used, the solution will be more stable than the solution using the Gaussian method
Cramer's_rule
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)
Calculations where numbers' precision is only limited by computer memory
for bignums, and others have libraries available for arbitrary-precision integer and floating-point math. Rather than storing values as a fixed number of
Arbitrary-precision arithmetic
Arbitrary-precision_arithmetic
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
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)
Logic-based number-placement puzzle
Guru Will Shortz, Time Magazine, March 2, 2009 Triangular Numbers, Gaussian Integers, and KenKen, The College Mathematics Journal, Volume 43, Number
KenKen
Representation theory of groups
L and K by the rings of algebraic integers they contain. One can see already in the case of the Gaussian integers that such bases may not exist: a +
Regular_representation
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
Graph operation
the square lattice over the Gaussian integers, and the 3-regular case uses triangular lattice over the Eisenstein integers. For convenience, an alternate
Goldberg–Coxeter_construction
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
travel, tourism, insurance
GAUSSIAN INTEGER
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