Successive divisions

In Arithmetic modular, the tests of divisions are the simplest technique and easiest to include/understand the algorithms of Factorization of integers.

That is to say a made up whole given N (in this article, N wants to say “the entirety to be factorized”), the tests of divisions consists in trying to divide N by each Prime number lower or equal to $\ sqrt {N}$ . If a number is found and that it is divided equally into N , a factor of N was found.

The tests of divisions are ensured to find a factor of N , as one checks all the factors first possible of N . In this way, if the algorithm fails, it is the proof that N is first.

The tests of divisions can be optimized in some ways. For example, if the last figure of N is not 0 or 5, the algorithm can avoid the test of factor 5. The same argument can be applied to 2 per checking from the last figure, and 3 by the checking of the sum of the figures. For more precise details, to see the criteria of divisibility.

In the Worse of the cases, the tests of divisions are an ineffective algorithm. If it starts from 2 and finishes with the square Racine of N , the algorithm requires

$\ pi (\ sqrt {N})$

tests of divisions, where π ( X ) is the quantity of prime numbers lower than '' X ''. This does not take account of wide Test of primality. If an alternative is used without test of primality, but simply by dividing each odd number lower than the square root of N , first or not, it takes ${\ sqrt {N} \ over 2}$ tests of divisions. This wants to say that for a N with great factors first of a similar size (like those used for the keys of asymmetrical Cryptographie), to factorize N by tests of division requires a crippling number of operations.

Nevertheless, for N with at least a small factor, the tests of divisions can be a fast manner to find this small factor. It is interesting to note that for a random N , there exists 50% of chances that 2 is a factor of N , and 33% of chances that 3 is a factor, and so on. It can be shown that 88% of all the positive entireties have a factor lower than 100, and that 91% have a factor lower than 1.000.

For more significant factorizations, nevertheless, others algorithms are more efficient and consequently feasible.

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