Integer Factorization: Meet the MetaConstellation


I am so in the weeds with this, now. LOL

The MetaConstellation … so far

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.

The Constellation for 321

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.)

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