Quadratic sieveThe quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve. It is a general-purpose factorization algorithm, meaning that its running time depends solely on the size of the integer to be factored, and not on special structure or properties.
Sieve of AtkinIn mathematics, the sieve of Atkin is a modern algorithm for finding all prime numbers up to a specified integer. Compared with the ancient sieve of Eratosthenes, which marks off multiples of primes, the sieve of Atkin does some preliminary work and then marks off multiples of squares of primes, thus achieving a better theoretical asymptotic complexity. It was created in 2003 by A. O. L. Atkin and Daniel J. Bernstein. In the algorithm: All remainders are modulo-sixty remainders (divide the number by 60 and return the remainder).
Quadratic residueIn number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.
Sieve theorySieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The prototypical example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the prototypical example of a sieve is the sieve of Eratosthenes, or the more general Legendre sieve. The direct attack on prime numbers using these methods soon reaches apparently insuperable obstacles, in the way of the accumulation of error terms.
General number field sieveIn number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10100. Heuristically, its complexity for factoring an integer n (consisting of ⌊log2 n⌋ + 1 bits) is of the form in O and L-notations. It is a generalization of the special number field sieve: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from prime powers (which are trivial to factor by taking roots).
Sieve of EratosthenesIn mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime number, 2. The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between them that is equal to that prime. This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime.
Quadratic fieldIn algebraic number theory, a quadratic field is an algebraic number field of degree two over , the rational numbers. Every such quadratic field is some where is a (uniquely defined) square-free integer different from and . If , the corresponding quadratic field is called a real quadratic field, and, if , it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a subfield of the field of the real numbers.
Sieve of SundaramIn mathematics, the sieve of Sundaram is a variant of the sieve of Eratosthenes, a simple deterministic algorithm for finding all the prime numbers up to a specified integer. It was discovered by Indian student S. P. Sundaram in 1934. Start with a list of the integers from 1 to n. From this list, remove all numbers of the form i + j + 2ij where: The remaining numbers are doubled and incremented by one, giving a list of the odd prime numbers (i.e., all primes except 2) below 2n + 2.
Quadratic formIn mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K. If , and the quadratic form equals zero only when all variables are simultaneously zero, then it is a definite quadratic form; otherwise it is an isotropic quadratic form.
Quadratic integerIn number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form x2 + bx + c = 0 with b and c (usual) integers. When algebraic integers are considered, the usual integers are often called rational integers. Common examples of quadratic integers are the square roots of rational integers, such as , and the complex number i = , which generates the Gaussian integers.
Quadratic reciprocityIn number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: Let p and q be distinct odd prime numbers, and define the Legendre symbol as: Then: This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form for an odd prime ; that is, to determine the "perfect squares" modulo .
Ideal class groupIn number theory, the ideal class group (or class group) of an algebraic number field K is the quotient group JK/PK where JK is the group of fractional ideals of the ring of integers of K, and PK is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of K. The order of the group, which is finite, is called the class number of K. The theory extends to Dedekind domains and their field of fractions, for which the multiplicative properties are intimately tied to the structure of the class group.
Factor baseIn computational number theory, a factor base is a small set of prime numbers commonly used as a mathematical tool in algorithms involving extensive sieving for potential factors of a given integer. A factor base is a relatively small set of distinct prime numbers P, sometimes together with -1. Say we want to factorize an integer n. We generate, in some way, a large number of integer pairs (x, y) for which , , and can be completely factorized over the chosen factor base—that is, all their prime factors are in P.
Quadratic equationIn algebra, a quadratic equation () is any equation that can be rearranged in standard form as where x represents an unknown value, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.) The numbers a, b, and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term.
Quadratic irrational numberIn mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients.
Binary quadratic formIn mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables where a, b, c are the coefficients. When the coefficients can be arbitrary complex numbers, most results are not specific to the case of two variables, so they are described in quadratic form. A quadratic form with integer coefficients is called an integral binary quadratic form, often abbreviated to binary quadratic form. This article is entirely devoted to integral binary quadratic forms.
Class number problemIn mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields (for negative integers d) having class number n. It is named after Carl Friedrich Gauss. It can also be stated in terms of discriminants. There are related questions for real quadratic fields and for the behavior as .
Isotropic quadratic formIn mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More explicitly, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if q(v) = 0. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form. Suppose that (V, q) is quadratic space and W is a subspace of V.
