Truly I find myself in the weeds here, folks. And that’s a good thing.
It’s good because the process of getting myself out of the tangle of weeds could involve untangling the principal tangle with which I’ve concerned myself for this past almost-year.
I wrote the Perl script for the Initial Trapezoid I talked about yesterday, and have run it for some sample values. What I’ve done with the scribbles here is run the script to get the x values for the top and bottom of the Initial Trapezoids for consecutive numbers in the remainder sequence produced by f(x) = x2+2cx+r, x=0,1,2,…, for an odd positive integer m=c2-r that one wishes to factor. Here, I did it for m=183.
I don’t understand these numbers yet, not enough either to discard them as not being useful for further insight, or to derive that further insight from them. I am looking for a way to examine the Initial Trapezoid for the beginning of a remainder sequence, or a small number of subsequent trapezoids, to determine with a small (hopefully fixed) number of iterations when f(x) yields a perfect square, therefore giving the least amount to add to c so that (c+x)2-m yields a perfect square and we can perform Fermat factorization by noting s2-t2=(s+t)(s-t).
And I’m actually happy not to understand the numbers I’m seeing. More good math study must happen. And if the MIRIAM process (Moving Inspiration Rapidly Into Accepting Minds) stays active – as it has done – that work will happen, and I will happily share it here. I still feel a strong sense of purpose and pattern in my pursuit of this. That’s a good feeling.