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

either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime. The factorizations take the form of an optional unit multiplied...

method for integers Fermat's factorization method for integers Monoid factorisation Multiplicative partition Table of Gaussian integer factorizations Hardy;...

factor Factorization Euler's factorization method Integer factorization Program synthesis Table of Gaussian integer factorizations Unique factorization Lehman...

combinatorics, a 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...

The unique factorization of Euclidean domains is useful in many applications. For example, the unique factorization of the Gaussian integers is convenient...

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

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

theory, an aurifeuillean factorization, named after Léon-François-Antoine Aurifeuille, is factorization of certain integer values of the cyclotomic polynomials...

In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers...

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...

the reverse of 79, 97, is also a prime. A Fortunate prime. A circular prime. A prime number that is also a Gaussian prime (since it is of the form 4n...

the integral definition of the gamma function, resulting in a Gaussian integral. In general, for non-negative integer values of n {\displaystyle n} we...

the equality of their digit sums with the digit sums of their prime factorizations. Arithmetic dynamics Casting out nines Checksum Digital root Hamming...

imaginary part and real part of the form 3 n − 1 {\displaystyle 3n-1} ; a Gaussian prime with no imaginary part and real part of the form 4 n − 1 {\displaystyle...

application of quantum computation is for attacks on cryptographic systems that are currently in use. Integer factorization, which underpins the security of public...

point-of-view. In particular, Linsker showed that if s {\displaystyle \mathbf {s} } is Gaussian and n {\displaystyle \mathbf {n} } is Gaussian noise with...

373, 379, 383, 397 (OEIS: A046066) Prime elements of the Gaussian integers; equivalently, primes of the form 4n + 3. 3, 7, 11, 19, 23, 31, 43, 47, 59...

isolated prime, a Chen prime, a Gaussian prime, a safe prime, and an Eisenstein prime with no imaginary part and a real part of the form 3 n − 1 {\displaystyle...

Schönhage–Strassen algorithm for fast multiplication of integers and polynomials. Integer factorization algorithms include the Elliptic Curve Method, the...

Gaussian integers, saying that it is a corollary of the biquadratic law in Z [ i ] , {\displaystyle \mathbb {Z} [i],} but did not provide a proof of either...

conjecture on the relation between the order of the center of the Steinberg group of the ring of integers of a number field to the field's Dedekind zeta...

elaboration of the concept of field. They are numbers that can be written as fractions a/b, where a and b are integers, and b ≠ 0. The additive inverse of such...

fundamental theorem of arithmetic, which says that every positive integer can be factored uniquely into prime numbers. Unique factorizations do not always exist...

theory, an integer q is a quadratic residue modulo n if it is congruent to a perfect square modulo n; that is, if there exists an integer x such that...

number that is 1 mod 4, the ideal ⁠ ( p ) {\displaystyle (p)} ⁠ in the Gaussian integers Z [ i ] {\displaystyle \mathbb {Z} [i]} is a Carmichael ideal. Both...

definition of sub-exponential time. An example of such a sub-exponential time algorithm is the best-known classical algorithm for integer factorization, the...

a Gaussian random polytope. A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges, and in fact, faces of all...

all entries remain integers if the initial matrix has integer entries Tridiagonal matrix algorithm — simplified form of Gaussian elimination for tridiagonal...

Gauss introduced the ring of Gaussian integers Z [ i ] {\displaystyle \mathbb {Z} [i]} , showed that it is a unique factorization domain, and generalized...