number theory, quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it is a...

square-free integer different from 0 {\displaystyle 0} and 1 {\displaystyle 1} . If d > 0 {\displaystyle d>0} , the corresponding quadratic field is called...

quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers...

Gaussian integers do not have a total order that respects arithmetic. Gaussian integers are algebraic integers and form the simplest ring of quadratic integers...

rounding-to-integer functions. The reason this satisfies N(ρ) < N(β), while the analogous procedure fails for most other quadratic integer rings, is as...

Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks...

unity) is a quadratic integer. For n = 5, 10, none of the non-real roots of unity (which satisfy a quartic equation) is a quadratic integer, but the sum...

The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field...

ideals. The quadratic integer rings are helpful to illustrate Euclidean domains. Quadratic integers are generalizations of the Gaussian integers in which...

{\frac {1}{2}}(1+{\sqrt {d}}\,)} respectively. See Quadratic integer for more. The ring of integers of the field F = Q [ α ] {\displaystyle F=\mathbb {Q}...

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...

Ramanujan–Nagell equation. 7 is one of seven numbers in the positive definite quadratic integer matrix representative of all odd numbers: {1, 3, 5, 7, 11, 15, 33}...

quadratic form on a vector space. The study of quadratic forms, in particular the question of whether a given integer can be the value of a quadratic...

decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater...

in quadratic form. A quadratic form with integer coefficients is called an integral binary quadratic form, often abbreviated to binary quadratic form...

rings of quadratic integers. In summary, if O d {\displaystyle {\mathcal {O}}_{\sqrt {d}}} is the ring of algebraic integers in the quadratic field, then...

{\displaystyle d} is a square-free integer and K = Q ( d ) {\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,)} is the corresponding quadratic field, then O K {\displaystyle...

totient function. Quadratic residue: An integer a is a quadratic residue modulo m, if there exists an integer x such that x2 ≡ a (mod m). Euler's criterion...

constitute the multiplicative group of integers modulo n. In the ring Z[√3] obtained by adjoining the quadratic integer √3 to Z, one has (2 + √3)(2 − √3) =...

case, integer programming problems are in many practical situations (those with bounded variables) NP-hard. 0–1 integer programming or binary integer programming...

unknown, it is called "univariate". The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation...

martingales Quadratic reciprocity, a theorem from number theory Quadratic residue, an integer that is a square modulo n Quadratic sieve, a modern integer factorization...

Hence, any 0–1 integer program (in which all variables have to be either 0 or 1) can be formulated as a quadratically constrained quadratic program. Since...

algebraic numbers. If the quadratic polynomial is monic (a = 1), the roots are further qualified as quadratic integers. Gaussian integers, complex numbers a...

major use in the formula for solutions of a quadratic equation. Quadratic fields and rings of quadratic integers, which are based on square roots, are important...

binary quadratic forms is isomorphic to the narrow class group of Q ( d ) {\displaystyle \mathbb {Q} ({\sqrt {d}})} . For real quadratic integer rings...

concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a...

real-valued function of an integer or natural number variable). Examples of quadratic growth include: Any quadratic polynomial. Certain integer sequences such as...

quadratic reciprocity—Let p and q be distinct odd prime numbers, and define the Legendre symbol as ( q p ) = { 1 if  n 2 ≡ q mod p  for some integer ...

and applied them to quadratic, cubic, and biquadratic reciprocity laws. For an odd prime number p and an integer a, the quadratic Gauss sum g(a; p) is...