In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition...

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 +...

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...

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...

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...

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...

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...

rational integers, such as 2 {\textstyle {\sqrt {2}}} , and the complex number i = − 1 {\textstyle i={\sqrt {-1}}} , which generates the Gaussian integers. Another...

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...

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...

In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued...

= 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...

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...

are often called the "rational integers" because of this. The next simplest example is the ring of Gaussian integers Z [ i ] {\displaystyle \mathbb {Z}...

algorithm Gaussian brackets – described on WolframMathWorld Gaussian's modular arithmetic Gaussian integer, usually written as Z[i] Gaussian prime Gaussian logarithms...

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...

without using quartic reciprocity. For an odd Gaussian prime π {\displaystyle \pi } and a Gaussian integer α {\displaystyle \alpha } relatively prime to...

In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that...

In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of...

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...

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...

optics, a Gaussian beam is an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function;...

integers. Its prime elements are known as Gaussian primes. Not every number that is prime among the integers remains prime in the Gaussian integers;...

reporting many computations – especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as...

of "integers" on complex numbers instead of real numbers, like Gaussian integers and Eisenstein integers. If we regard the ring of Gaussian integers, we...

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...

second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas. He said that a third monograph...

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...

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}}}...

In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs...