Given a semiprime: $N = 688307 = 431 * 1597$
$ceil(sqrt(N)) = 830$
$830^2 - N = 593$
$n^2 + 1660n − k^2 + 593 = 0$
$n = 184, k = 583$
$p1 = (830 + n) - k = 1014 - 583 = 431$
$p2 = (830 + n) + k = 1014 + 583 = 1597$
First, I tried an approach to express $k$ as a approximation of some power function (e.g. $Ax^B$), but I guess this doesn't make sense. The analytical form of $k$ is just as simple as:
$k = \sqrt{n^2+1660n+593}$
but this means nothing useful, since subtitition will give identity equation.
Could, however analyzing fractional parts of squre root for each iteration of Fermat factorization give anything useful? Looking into first 400 iterations:
This looks pretty chaotic, especially the top left part, but I guess some clarity may be given by expressing another function, that returns the difference from previous iteration:
Obviously, I'm not the first who found this, but is this something that can facilitate to find the factors quicker that in "naive" implementations of Fermat factorization?
I do not believe that. The key problem is to find a representation of $N$ as the difference between two squares.
$(\frac{1597+431}{2})^2 - N = (\frac{1597-431}{2})^2$
$(1014)^2 - N = (583)^2$
$(830+184)^2 - N = (583)^2$
There is a nice visualization of Fermat's factorization in the general divisor plane
Annotation: In the general divisor plane, each divisor point describes a solution to the divisor relation $x|y$.
Unfortunately, the fractional parts of the solution candidates say nothing about the 'divisor point neighborhood'.