Fermat method
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 $ij$.

4.
Increment iterator $i$ once. If $$, 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, $152=13$ 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.
References
 1 R. Crandall & C. Pomerance, Prime Numbers^{}: A Computational Perspective, Springer, NY, 2001: 5.1
Title  Fermat method 

Canonical name  FermatMethod 
Date of creation  20130322 16:39:03 
Last modified on  20130322 16:39:03 
Owner  PrimeFan (13766) 
Last modified by  PrimeFan (13766) 
Numerical id  6 
Author  PrimeFan (13766) 
Entry type  Algorithm 
Classification  msc 11A41 
Synonym  Fermat’s method 
Synonym  Fermat factorization method 
Synonym  Fermat’s factorization method 