Gourt > Science > Math > Number Theory > Factoring (3)
- This article is about the mathematical concept. For the financial term see factoring (finance).
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 all 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.
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Computing Aurifeuillian factors - Algorithm that finds non-trivial factors of certain numbers of the form a^b +/- 1.
Discovery of a Lost Factoring Machine - Built by a French amateur, E.-O. Carissan, around 1919. Shallit, Williams and Morain include photographs and references to their paper.
Factoring Fermat Numbers - Cash prizes for new factors of Fermat numbers Fn, for n = 12 through 22.
Factoring Papers - Links to papers on the theory and practice of factoring.
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Factoris - Online calculator that factorizes large numbers, specified by formula.
Meta Description: [ factors integers and polynomials. This is the main site of WIMS (WWW Interactive Mathematics Server): interactive exercises, online calculators and plotters, mathematical recreation and games ]
Factorization of F10 - F10 = 2^(2^10) + 1 is the 10-th Fermat number. Richard Brent describes his discovery of the two largest factors.
Factorization of RSA-155 - Announcement of factorization of a 512-bit RSA key using the General Number Field Sieve (GNFS).
Factorization Using the Elliptic Curve Method - Java applet that can be used to find 20- or 30-digit factors of numbers or numerical expressions up to 1000 digits long. It also computes the number and sum of divisors, the Euler's totient and moebius functions, and the decomposition of the number in a sum of up to four squares.
Meta Description: [ Applet that can be used to find 20- or 30-digit factors of numbers up to 10000 digits long ]
Factorizations of Near-repdigit Numbers - Factorizations of numbers composed of all the same digit except first and/or last.
FactorWorld - Dedicated to algorithms and computational results on integer factorization. Includes links to papers, downloadable software, and online resources.
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Fermat Search - Leonid Durman's Fermat number factoring site and program.
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Generalised Mersenne Numbers - Numbers of the form b^n +- (b-1), factorised by Wojciech Florek.
Integer Factorization - Paul Leyland's list of his own and other factorisations.
Known Amicable Pairs - A listing of all the known pairs of numbers, each of which is the sum of the aliquot divisors of the other. Complete for smaller numbers, and extending beyond 200 digits.
N!+-1 Factoring Status - Factoring efforts that have been made so far on numbers of the form n!+-1 using ECM factoring.
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Number Field Sieve - Triade systems links to papers on the number field sieve.
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Paul Herman's Factoring Page - A thorough summary of many major factoring methods. Includes some source code on many pages, gentle introductions to the more complex methods and further links.
Robinson Primes - An analysis of problems relating to the numbers k.2^n+-1, primes, and factor patterns, including the Sierpinski problem.
RSA Laboratories Factoring Challenge - Numbers representative of those used in the RSA cryptosystem are offered for factor attempts with prizes. A Partition List challenge is also provided in order to encourage work on factoring in general.
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Sam Wagstaff - Article about SNFS factorisation of 3^349-1.
Sierpinski Problem - Sierpinski proved there exist infinitely many odd integers k such that k*2^n+1 is composite for every n. Ray Ballinger coordinates a search to prove or disprove whether k=78557 is the smallest solution.
The Anti-Divisor - A definition and description of the Anti-Divisor, and some related results.
The Factor Zone - Aimed at grade school students and teachers, includes course guidelines, worksheets, and factor tables up to 600.
The XYYXF Project - A collaborative project to produce the factorizations of x^y + y^x for 1yx101.
Meta Description: [ A collaborative project to find the factorizations of x^y+y^x for 1
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