Question on factoring by the difference in similar polynomials
- I have been experimenting for a while with factoring by the difference
in polynomials such that:
Specifically looking for cases where ((x^2+x)/z) - an = (y^2+y)/z
Assuming a brute force approach by choosing a value for z, solving for
X^2+x-zan and then x=|x|+1, for sequential values of a, then it seems
that the density of solutions increases for increasing z, whilst the
required search area decreases such that the maximum value of a
Now one of my questions is: Would there be an optimum value of z where
the solutions would be quite dense as compared to the search area
(limited to n/z). Thus making it a serious factoring method.
The second question would be is there any way to apply Kraitchik's
method to constructs of (x^2+x)/z rather than just x^2.
The reason that I was interested in this method was that it seemed
that by carefully choosing values, for z and a, that smallish residues
could be obtained and the law of small numbers could start coming into
play and give an even higher than expected density of solutions.