Fermat's Factorization Method

Given a number , Fermat's factorization methods look for integers and such that . Then

and is factored. A modified form of this observation leads to Dixon's factorization method and the quadratic sieve.

Every positive odd integer can be represented in the form by writing (with ) and noting that this gives

Adding and subtracting,

so solving for and gives

Therefore,

As the first trial for , try , where is the ceiling function. Then check if

is a square number. There are only 22 combinations of the last two digits which a square number can assume, so most combinations can be eliminated. If is not a square number, then try

so

Continue with

so subsequent differences are obtained simply by adding two.

Maurice Kraitchik sped up the algorithm by looking for and satisfying

i.e., . This congruence has uninteresting solutions and interesting solutions . It turns out that if is odd and divisible by at least two different primes, then at least half of the solutions to with relatively prime to are interesting. For such solutions, is neither nor 1 and is therefore a nontrivial factor of (Pomerance 1996). This algorithm can be used to prove primality, but is not practical. In 1931, Lehmer and Powers discovered how to search for such pairs using continued fractions. This method was improved by Morrison and Brillhart (1975) into the continued fraction factorization algorithm, which was the fastest algorithm in use before the quadratic sieve factorization method was developed.