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Number Theory

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Factorization
  Prime Factorization Trial Division
  Fermat Factorization
  Pollard Rho Factorization

Factorization

Factorization is one of the methods to verify whether a number is prime.

Prime Factorization Trial Division

The 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 non-trivial prime divisors.

Let x, y are non-trivial factors of integer N with N=xy. imply

image

It is not necessary to test all numbers from 1 to N-1.

If x, y are non-trivial 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, x|N, 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 non-trivial prime factor.

Fermat Factorization

The 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

image

Since all even number can be divided by 2, let x, y are non-trivial odd factors of integer N with N=xy. imply

image

Equate two equations. Imply

image

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

image

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, (a-b) and (a+y) are the non trivial factors of N. Imply

image

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 Factorization

The 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 pseudo-random sequence to increase the probability of getting prime factors of the composite number.


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ID: 120400011 Last Updated: 17/4/2012 Revision: 0 Ref:

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References

  1. R. Paulo, 1996, The New Book of Prime Number Records
  2. Wolstenholme, R.J., 1862, On Certain Properties of Prime Numbers
  3. Mann, H.B., Shanks D., 1972, A Necessary and Sufficient Condition for Primality, and Its Source
  4. J.M Pollard, Kangaroos, 1975, A Monte Carlo Method for Factorization
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