I am finally barking up the correct tree, so to speak, and I have found the little squirrel I sought.

From the beginning, with gusto! (By the way, the relationship I am about to describe took a while to hammer out on paper, and a great deal longer to get right in Perl code.)

The task is the factorization of an odd integer m, which is the product of two distinct prime factors p and q (also both odd), and considered in the order such that p < q. Formulaically, m = pq.

Let s be the smallest integer greater than the square root of m. s is positive since p and q are distinct. Let r = s^{2}-m. Now consider the positive integers c_{1} and c_{2} which satisfy p = s-c_{1} and q = s+c_{2}. Let n = c_{2}-c_{1}.

From observing a large amount of data, I determined a relationship between r and the constant n as follows:

Let a = (q-p)^{2}/4, which will always be an integer. Let b = n/2.

r = ab(p+q-b).

I am not at all certain of the likelihood of success, but for now, my approach to the integer factorization problem boils down to using the value of r = m – s^{2} and the relationship above to speed up the search for p and q. A place to start is to note that:

p+q = b+r/(ab). It may be a trivial relationship, due to the nature of the values of c_{1} and c_{2}; I have not yet determined whether this is so.

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