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In mathematics, factorization or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5; and the polynomial x2 − 4 factors as (x − 2)(x + 2). In both cases, we obtain a product of simpler things.The aim of factoring is usually to reduce something to "basic building blocks", such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra.
Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.
A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.
1 Factoring in mathematical logic
In mathematical logicMathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. Although the layperson may think that mathematical log and automated theorem provingProofs Automated theorem proving (the currently most important subfield of automated reasoning is the proving of mathematical theorems by a computer program. Depending on the underlying logic, the problem of deciding the validity of a theorem varies from, factoring is the technique of deriving a single, more specific atomFor alternative meanings see atom (disambiguation). An atom is a microscopic structure found in all ordinary matter around us. Atoms are composed of 3 types of subatomic particles: electrons, which have a negative charge; protons, which have a positive ch from a disjunction of two more general unifiableSee also: Unification Church Unification can also refer to the German reunification of East and West Germany. In mathematical logic, in particular as applied to computer science, a unification of two terms is a join (in the lattice sense) with respect to atoms. For example, from ∀ X, Y : P(X, a) or P(b, Y) we can derive P(b, a).
2 See also
Prime factorization algorithmA prime factorization algorithm is an algorithm (a step-by-step process) by which an integer (whole number) is "decomposed" into a product of factors that are prime numbers. The fundamental theorem of arithmetic guarantees that this decomposition is uniqu, program synthesisProgram synthesis comprises a range of technologies for the automatic generation of executable computer programs from high-level specifications of their behaviour. In contrast to compilation, the specifications are usually non- algorithmic. The idea origi
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