What you see above is a graph whose shape and jagged characteristics are similar to those of graphs I have been examining for two years, though they have shown up in different stages in my study and growing understanding of the problem of fast integer factorization.

Most recently, my quickfact.pl Perl script and the mathematics behind it center on looking at the gaps between the first and third elements of the Remainder Series x^{2}+2cx+r for x=0,1,2,… and c and r being the Ceiling Root and Remainder for odd positive integer m=c^{2}-r, and pairs of consecutive even or odd perfect squares greater than those values. I have discovered, using my Zone Grid construction, that the zone boundary corresponding to a factorization solution is the first zone boundary where the difference in the two gaps evenly divides the size of the gaps themselves. This ensures, to state it briefly and without adequate explanation, that the whole series of gaps formed by extending the Remainder Series downward along with the series of perfect squares, will hit exactly 0, and will do so at a point where the perfect square corresponding to that 0 value will be the one which, when added to m, produces a larger perfect square and thus permits Fermat factorization of m.

The jagginess of this graph, of this series of “gapdiff” values, is what makes coming up with a formulaic way to jump straight to the solution so difficult. But the jagginess is systematic: It appears similar, in my mind, to taking a quadratic curve (parabola) and modifying its values modulo positive integer values on a straight line. That in fact is what I perceive might be happening with the Zone Grid boundaries and the values on the Remainder Line (a quadratic curve rendered straight by the Zone Grid’s properties). I just need to figure out how to undo it.

Notice, for instance, how the jagged green curve (the delta-one of the gapdiff series) seems to have mostly straight plateaus, the first of which suddenly drops to the second when it gets to a y-value a little greater than 150, and then drops again when it gets to a y-value a little greater than 75. This is the sort of pattern that calls out for me to experiment with other numbers and look for similar behavior – and then, of course, to put the hard work into figuring out what that all means and how I can use it.

That is what’s so exciting to me: the possibility that all of these concepts I’ve constructed will allow this intractable kinkiness to become calculable: to permit me, given c and r, to determine s and t such that m=s^{2}-t^{2}.

In other words, just when I think I’m done for now, inspiration beckons me to close the gap between what I know and the answers I seek.

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