Integer Factorization: Patterns in the Weeds


When m is an odd positive integer, c2 is the least perfect square greater than or equal to m that is odd or even as (m+1)/2 is odd or even, and r=c2-m, we can construct a Zone Grid and compute the slope of a Remainder Line passing through appropriately chosen values of r0=r and r1=(c+1)2-m, which also happen to be the x=0 and x=1 values of f(x)=x2+2cx+r. Since the Zone Grid provides a method for arranging integer values of quadratic polynomials in straight lines, we then plot the intersections between the Remainder Line and “zone boundaries” demarcated by the straight lines composed only of perfect squares.

Now in order for this to be useful in factorization, in my study I have needed to look at how these numbers behave for small discrete semiprimes, but without putting too much stock in my knowing the answer to the question of how to factor them. In fact, my MacBook Pro does a pretty rapid job, using my script, of handling integers even at the ordinary integer data type limit for Perl. So I can know the answer pretty quickly, but I have to walk the line between ways I find knowing the answer to be useful, and ways in which knowing the answer is information I should not use in extracting patterns I wish the algorithm to find when the numbers do become unwieldy – or, more precisely, in the general case of an arbitrary-sized odd integer m.

I am hampered in this slightly by the fact that I do not yet have installed on my MacBook Pro, or in my mental programming toolkit, the means to work with arbitrary-precision arithmetic. But a pattern that will work with all arbitrary-sized integers will work with all integers of fixed size, and I am thinking perhaps that pattern will make itself recognizable to me without me engaging currently inaccessibly huge numbers to get to it.

I am looking at the ratios of the distances between r0 and the first zone boundary, and the first and second zone boundaries, along the Remainder Line. I also have available, but have not yet looked at, the position of r1 on the Zone Grid and Remainder Line as well. It gives me a set of distance ratios with which I can fiddle, but I am not out of the weeds (woods?) yet in finding a correspondence with the Ascent, the difference between (p+q)/2 and c for discrete semiprime m=pq. That’s the primary case in all of this where having the answer ahead of time might be useful. But it’s only useful if it’s useful, and that usefulness is currently deep in the weeds/woods. And during this whole year of study, I’ve been particularly adept at barking up trees whose revelations have only been profitable to my own insights on which tree to sniff next.

LOL, what tortured metaphors.

Here, finally, are a graph of Ascent vs. the distance ratios described in the above paragraph, and a graph of those two distances as the x/y points themselves, for the seventy or eighty of the smallest discrete semiprimes of non-zero Ascent I’ve examined so far :

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