Draft for Information Only Content
Factorization
FactorizationFactorization is one of the methods to verify whether a number is prime. Prime Factorization Trial DivisionThe direct search factorization is a bruce force factorization method for determining all possible prime factors of a composite number, N, through the repeated examination of divisibility by trial division from a set of nontrivial prime divisors. Let x, y are nontrivial factors of integer N with N=xy. imply It is not necessary to test all numbers from 1 to N1. If x, y are nontrivial factors of integer N with N=xy and x≤y, then x≤√N. Since if x>√N, then y≥x>√N and imply xy>N, which contradicts to the assumption N=xy. Therefore, the trial division can be performed by checking whether x divides N, xN, for x=2 to the floor of √N. if x is found, imply x≡0 (mod N), and y, y=N/x is a factor also. In verifying whether a number N is prime, x can be limited to a nontrivial prime factor. Fermat FactorizationThe Fermat factorization (1600s) from Pierre de Fermat is another way of factoring a composite number by considering the composite number as the difference of squares. This standard binomial can then be factorized into a product. Imply Since all even number can be divided by 2, let x, y are nontrivial odd factors of integer N with N=xy. imply Equate two equations. Imply Both a and b are integers because x and y are odd integers, the sum and difference between any two odd number are even number which is divisible by 2. As x and y are non trivial factors, √N≥x>1 and y≥x, imply a≥1 and b≥0. Instead of testing the non trivial factors x, and y, Fermat factorization examine the integers a and b. Imply Therefore, the Fermat factorization can be performed by checking a from the floor of √N to N. whether the corresponding value of b is an integer and the corresponding terms of the product, (ab) and (a+y) are the non trivial factors of N. Imply In verifying whether a number N is prime, the value of integer a should be choosen from the floor of √N to N. Pollard Rho FactorizationThe Pollard rho factorization or Pollard's Monte Carlo factorization method (1975) from J.M. Pollard is another technique of finding factors of a composite number by making use of probabilistic ideas from transforming a sequencial sequence to a congruential pseudorandom sequence to increase the probability of getting prime factors of the composite number. ©sideway References
ID: 120400011 Last Updated: 2012/4/17 Revision: Ref: 
Home (1) Computer Hardware (148) Software Application (187) Digitization (24) Numeric (19) Programming Web (540) CSS (SC) HTML Knowledge Base Common Color (SC) Html 401 Special (SC) OS (367) MS Windows Windows10 (SC) DeskTop (5) Knowledge Mathematics Formulas (8) Number Theory (206) Algebra (17) Trigonometry (18) Geometry (18) Calculus (67) Complex Analysis (5) Engineering Tables (8) Mechanical Mechanics (1) Rigid Bodies Statics & Dynamics (129) Fluid (5) Fluid Kinematics (5) Control Process Control (1) Acoustics (19) FiniteElement (2) Biology (1) Geography (1) 
Latest Updated Links

Copyright © 20002019 Sideway . All rights reserved Disclaimers last modified on 10 Feb 2019