Integer Factorization: This Morning’s Delivery


It is a rainy Saturday of Holy Week, 2023, smack-dab between Good Friday and Easter/Resurrection Sunday on the religious calendar which I personally celebrate.

It is also a year, give or take a week, since inspiration started coming to me in the early morning hours, usually after our chugs woke me up to have an early potty walk, with integer factorization ideas keeping me from returning to slumber.

My wife was away visiting grandchildren then. She is here now, upstairs, not yet asleep. It is 5:54 a.m. That is how it goes.

And now, the ideas ring with that insistent resonance that keeps me awake again, compelling me to come downstairs from the bedroom and work on them.

So here we go, jumping right into it. For recent background, see here.

Also, if you’re not particularly math-y, feel free to skip the following paragraph. There’s also some metaphorical and philosophical stuff after that.

I just finished describing the concept of a Great Trapezoid associated with the Remainder Line of an odd positive integer m as placed systematically on the Zone Grid. I can construct such a Great Trapezoid using zone boundaries and Ascent lines (lines through constant x values) only if m has a non-trivial factorization. However, there is another simple figure, let’s call it the Great Rectangle, that one can always draw on the Zone Grid, simply the rectangle created by the initial point corresponding to m on the Grid and the point on the far end, so to speak, of the segment of the Remainder Line where the Trivial Ascent appears: i.e., corresponding to the value of ((m+1)/2)-c, where m is the odd integer we are considering and c is its Ceiling Root.

A metaphoric connection has jumped out at me to help keep me awake to pursue these ideas, like these inspirationss tend to do: a connection between these recent ideas and some that I had more than a year ago with other mathematical studies I was doing, studies in which I was inspired by ideas in Rebecca Newberger Goldstein‘s novel “36 Arguments for the Existence of God: A Work of Fiction.” I began thinking of prime numbers back then as “Azarya’s Angels,” based on one of her fictional characters and his idea that prime numbers were indeed angels. The connection now is that the odd prime numbers still do have Ascents associated with their trivial factorization, and thus have Rectangles on the Zone Grid, in all their bilateral symmetry, but not Grand Trapezoids, less symmetric and therefore not quite as perfect. Their Ascents are their ONLY meaningful point on the Zone Grid.

Philosophical and Metaphoric Tangents Abound: We living terrestrial beings are also not as manifestly perfect as the angels, even Azarya’s, our selves being composites of matter and spirit, or something we can consider to be this, maintained by body and brain, but not yet understood, not biologically, not neuro-scientifically, not even philosophically, so that we, like large numbers whose factorization nobody yet knows, don’t understand our non-Trivial ascent which of course would be more reachable for us than that far-distant Trivial (or Ultimate) Ascent we can calculate, when we join the Angels at the End of the Line.

More work, more motivation, more inspiration. What an anniversary gift! I hope to be able to give a gift as beautiful right back, before too long. But first I need to Ascend to it. And that’s non-trivial.

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