I am so in the weeds with this, now. LOL
Sometimes, when the MIRIAM process (Moving Inspiration Rapidly Into Accepting Minds) gives me a glimpse of somewhere in this integer factorization study that I can go, something that I can do to assemble an idea into reality … the process tells me NOTHING about what it means or really how to use it in the actual process of developing a better formula for factoring integers. Sometimes it’s just a property that shows up, a glimpse of beauty and something like-but-unlike symmetry, and it gives me a good bit of work to do, and a somehow satisfying result.
I was looking at discrete semiprimes, and focusing again on them in particular, starting with the least products of two distinct primes whose Ascents, which connect the ceiling-square-and-remainder formula to their Fermat factorizations. currently defy the development of a compact formula. I was looking at the Characteristic Polynomial f(x)=x2+2cx+r that accompanies the expression of the discrete semiprime m=c2-r and gives a hint of which Ascent will give factorization. Finally, inspired by my Zone Grid and Beehive Plot ideas from earlier, I was making charts of distances of the x=0 and x=1 values of the Characteristic Polynomials from pairs of successive perfect squares.
You know, just to see what would happen. 😀
When I started lining up, for a given discrete semiprime, parallel sets of measures of the gaps between the Characteristic Polynomial line and successive Zone Boundary lines, I ended up with rows of regularly spaced dots, going top-to-bottom in periods decreasing by two, until one of the rows hit the point corresponding to a gap of zero, which ties to the ever-elusive Fermat factorization answer. And that’s when that little tickle in my brain last night got me truly enthusiastic about this.
I looked at some of the constellations of these dots, marked below with an X, and I realized that I was seeing similar arrangements of dots in the constellations, so to speak, for different discrete semiprimes whose values also happened to be close to each other.
What I was starting to realize last night before bedtime, to my increasing delight and excitement, was that there was a sensibly small region I could imagine within which these constellations for these consecutive discrete semiprimes would fit together, to make a sort of constellation of their constellations: a Meta-Constellation.
The meaning and application of such an obvious but somewhat imprecisely defined process have yet to blossom in my mind, but the fact that the random-seeming behavior of these gaps and of the Ascent seems to concretize into a form that seems to have some pattern and beauty to it … well, that counts as something of a breakthrough, in my thinking. It’s progress on a problem that has puzzled humanity since we started asking the question “How do you factor a large number.”
I have not yet looked at the constellations for larger numbers yet. I wonder what those probably more immense arrangements will look like, where they will fit in relation to the small-number-based MetaConstellation pictured above, and what that can tell me.
Stay tuned, space travelers! Like I said, I’m down in the weeds with this, but weeds do grow in fertile ground, and can hide treasures there.
(Also dog poop, but never mind that.)