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[parent] Fermat method (Algorithm)

The Fermat method is a factorization algorithm for odd integers that tries to represent them as the difference of two square numbers.

Call the Fermat method algorithm with an integer $ n = 2x + 1$.

  1. Initialize the iterator $ i = \lceil \sqrt{n} \rceil$ and test cap at $ \sqrt{i^2 - n}$.
  2. Compute $ j = \sqrt{i^2 - n}$.
  3. Test if $ j$ is an integer. If it is, break out of subroutine and return $ i - j$.
  4. Increment iterator $ i$ once. If $ i < \frac{n + 9}{6}$, go to step 2.
  5. Return $ n$.

For example, $ n = 221$. The square root is approximately 15, so that's what we set our iterator's initial state to. The test cap is 39.

At $ i = 15$, we find that $ \sqrt{15^2 - 221} = 2$, clearly an integer.

Then, $ 15 - 2 = 13$ and $ 15 + 2 = 17$. By multiplication, we verify that $ 221 = 13 \cdot 17$, indeed. By trial division, this would have taken 15 test divisions.

However, there are integers for which trial division performs much better than the Fermat method, such as numbers of the form $ 3p$ where $ p$ is an odd prime greater than 3.

Bibliography

1
R. Crandall & C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001: 5.1



"Fermat method" is owned by PrimeFan.
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Other names:  Fermat's method, Fermat factorization method, Fermat's factorization method

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examples of the Fermat method on a few integers (Example) by PrimeFan
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Cross-references: prime, odd, divisions, trial division, multiplication, square root, iterator, integer, numbers, square, difference, represent, odd integers, algorithm
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This is version 3 of Fermat method, born on 2007-02-01, modified 2007-02-22.
Object id is 8855, canonical name is FermatMethod.
Accessed 1976 times total.

Classification:
AMS MSC: 11A41 (Number theory :: Elementary number theory :: Primes)
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