Shanks' square forms factorization is a method for integer factorization devised by Daniel Shanks as an improvement on Fermat's factorization method.
The success of Fermat's method depends on finding integers
is the integer to be factored.
An improvement (noticed by Kraitchik) is to look for integers
Finding a suitable pair
does not guarantee a factorization of
, and there is a good chance that the prime divisors of
are distributed between these two factors, so that calculation of the greatest common divisor of
will give a non-trivial factor of
A practical algorithm for finding pairs
was developed by Shanks, who named it Square Forms Factorization or SQUFOF.
The algorithm can be expressed in terms of continued fractions or in terms of quadratic forms.
Although there are now much more efficient factorization methods available, SQUFOF has the advantage that it is small enough to be implemented on a programmable calculator.
Shanks programmed it on an HP-65, made in 1974, which has storage for only nine digit numbers and allows only 100 steps/keystrokes of programming.
There are versions of the algorithm that use little memory and versions that store a list of values that run more quickly.
In 1858, the Czech mathematician Václav Šimerka used a method similar to SQUFOF to factor
[1] Note This version of the algorithm works on some examples but often gets stuck in a loop.
, the integer to be factored, which must be neither a prime number nor a perfect square, and a small positive integer,
Output: a non-trivial factor of
is a perfect square at some odd value of
Start the second phase (reverse cycle).
is the recently calculated value of
is the recently calculated value of
[citation needed] Then if
is a non-trivial factor of
[citation needed] Shanks' method has time complexity
[2] Stephen S. McMath wrote a more detailed discussion of the mathematics of Shanks' method, together with a proof of its correctness.
is a perfect square, so the first phase ends.
Below is an example of C function for performing SQUFOF factorization on unsigned integer not larger than 64 bits, without overflow of the transient operations.
[citation needed]