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  2. Factorization of polynomials - Wikipedia

    en.wikipedia.org/wiki/Factorization_of_polynomials

    If two or more factors of a polynomial are identical, then the polynomial is a multiple of the square of this factor. The multiple factor is also a factor of the polynomial's derivative (with respect to any of the variables, if several). For univariate polynomials, multiple factors are equivalent to multiple roots (over a suitable extension field).

  3. Fermat's factorization method - Wikipedia

    en.wikipedia.org/wiki/Fermat's_factorization_method

    Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: =. That difference is algebraically factorable as (+) (); if neither factor equals one, it is a proper factorization of N.

  4. Integer factorization - Wikipedia

    en.wikipedia.org/wiki/Integer_factorization

    Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer n using mental or pen-and-paper arithmetic, the simplest method is trial division : checking if the number is divisible by prime numbers 2 ...

  5. Pollard's rho algorithm - Wikipedia

    en.wikipedia.org/wiki/Pollard's_rho_algorithm

    Occasionally it may cause the algorithm to fail by introducing a repeated factor, for instance when ⁠ ⁠ is a square. But it then suffices to go back to the previous gcd term, where gcd ( z , n ) = 1 {\displaystyle \gcd(z,n)=1} , and use the regular ρ algorithm from there.

  6. Quadratic sieve - Wikipedia

    en.wikipedia.org/wiki/Quadratic_sieve

    The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field sieve).It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve.

  7. Trainer shares advice to keep your dog settled if you’re busy ...

    www.aol.com/trainer-shares-advice-keep-dog...

    And, as DeWillems does beforehand, ensure that your dog’s had plenty of water, been to the bathroom, and enjoyed some exercise before you ask them to settle on their bed. This way, your dog’s ...

  8. images.huffingtonpost.com

    images.huffingtonpost.com/2012-04-25-20111115CPF...

    STATE OF COLORADO Department of State 1700 Broadway Suite 200 Denver, CO 80290 Scott Gessler Secretary of State William A. Hobbs Deputy Secretary of State Main Number Administrati

  9. Euler's factorization method - Wikipedia

    en.wikipedia.org/wiki/Euler's_factorization_method

    Euler's factorization method is a technique for factoring a number by writing it as a sum of two squares in two different ways. For example the number 1000009 {\displaystyle 1000009} can be written as 1000 2 + 3 2 {\displaystyle 1000^{2}+3^{2}} or as 972 2 + 235 2 {\displaystyle 972^{2}+235^{2}} and Euler's method gives the factorization ...

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