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Topic: Factorization


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  Factorization using the Elliptic Curve Method
The final value must have 10000 or fewer digits, intermediate results must have 20000 or fewer digits and in the case of divisions, the dividend must be multiple of the divisor.
In order to do it, run the factorization in the first computer from curve 1, run it in the second computer from curve 10000, in the third computer from curve 20000, and so on.
When the number to be factorized is in the range 31-90 digits, after computing some curves in order to find small factors, the program switches to SIQS (if the checkbox located below the applet enables it), which is an algorithm that is much faster than ECM when the number has two large prime factors.
www.alpertron.com.ar /ECM.HTM   (618 words)

  
 Factoring Numbers
This accidental over-duplication of factors is another reason why the prime factorization is often best: it avoids counting any factor too many times.
So it's best to stick to the prime factorization, even if the problem doesn't require it, in order to avoid either omitting a factor or else over-duplicating one.
The nice thing about this upside-down division is that, when you're done, the prime factorization is the product of all the numbers around the outside.
www.purplemath.com /modules/factnumb.htm   (685 words)

  
  The Prime Glossary: wheel factorization
To see if a number is prime via trial division (or to find its prime factors), we divide by all of the primes less than (or equal to) its square root.
Rather than divide by just the primes, it is sometimes more practical to divide by 2, 3, and 5; then divide by all the numbers congruent to 1, 7, 11, 13, 17, 19, 23, and 29 modulo 30--again stopping when you reach the square root.
This type of factorization is called wheel factorization and the spokes are the list of integers prime to all of the primes we are using.
primes.utm.edu /glossary/page.php?sort=WheelFactorization   (488 words)

  
  Factorization - Encyclopedia, History, Geography and Biography
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.
Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra.
Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function.
www.arikah.com /encyclopedia/Factorisation   (1323 words)

  
  Factorization - Wikipedia, the free encyclopedia
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.
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.
en.wikipedia.org /wiki/Factorization   (1096 words)

  
 PlanetMath: integer factorization   (Site not responding. Last check: )
The factorization of a positive integer is unique (this is the fundamental theorem of arithmetic).
The term “factorization” is often used to refer to the actual process of determining the prime factors.
This is version 3 of integer factorization, born on 2007-02-02, modified 2007-02-11.
planetmath.org /encyclopedia/IntegerFactorization.html   (328 words)

  
 Integer factorization - Wikipedia, the free encyclopedia
In number theory, integer factorization is the process of breaking down a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer.
When the numbers are very large, no efficient integer factorization algorithm is published; a recent effort which factored a 200 digit number (RSA-200) took eighteen months and used over half a century of computer time.
Given the state of the art as of 2006, the hardest instances of these problems are those where the factors are two randomly-chosen prime numbers of about the same size.
en.wikipedia.org /wiki/Integer_factorization   (1188 words)

  
 Grade 5: Prime Factorization: Developing the Concept
Now that students can find the prime factorization for numbers that are familiar products, it is time for them to use their rules for divisibility to find the prime factorization of unfamiliar numbers.
Say: It was better to start this process with two factors like 10 and 24 or 20 and 12 than to take two factors like 2 and 120 or 3 and 80, because each of the previous numbers can be broken down and the end result will probably take fewer steps.
Turning the problem around and giving students the prime factorization of a number and asking them what they know about the number without multiplying it out is a good way to assess their understanding of the divisibility rules, the concept of factor, and multiplication in general.
www.eduplace.com /math/hmm/background/5/07/te_5_07_factors_develop.html   (647 words)

  
 5.2 LU Factorization
The basic idea is that an LU factorization is actually implemented as a class in the Math.h++ class library.
Once constructed, this LU factorization can be used for many things.
This approach can result in great time savings because calculating the LU factorization takes most of the time, but need be done only once.
www.roguewave.com /support/docs/hppdocs/mthug/5-2.html   (406 words)

  
 Some Factorization Methods
Below are links to some factorizations algorithms developed by me. There are simple algorithms, their cost is linear to the size of n.
I decided to make this documents after spending few weeks (months) trying to develop fast (log n time) factorization algorithm.
This was quite unusual work for me - I spent most time with pen and notebook than in front of computer.
www.4neurons.com /other/Factorization/Factorize.html   (122 words)

  
 Integer Factorization Source Code
I wrote this library when I finally managed to understand the ins and outs of factoring large integers using advanced sieving methods (it took years).
Intermediate results are saved on disk and factorizations can be restarted later.
The core code is organized as a library with a lightweight API that allows easy integration with other applications.
www.boo.net /~jasonp/qs.html   (179 words)

  
 Cholesky Factorization
The block-partitioned form of Cholesky factorization may be inferred inductively as follows.
A snapshot of the block Cholesky factorization algorithm in Figure 5 shows how the column panel
The factorization can be done by recursively applying the steps outlined above to the
www.netlib.org /utk/papers/factor/node9.html   (203 words)

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