## Integer Factorization: My Mycelial Mathematical Meanderings?

It’s fun when I find a metaphor from the world of fiction, especially from the world of popular fiction, and incorporate it into my thinking about something real (or, more accurately, practical) with which I am dealing, and then I discover that the metaphor was, or I could easily make it, a more appropriate fit than I’d imagined at first.

The pop culture reference in question is the Star Trek concept of using the Spore Drive to travel vast distances (and sometimes to parallel universes) in “jumps” using the Mycelial Network, a vast living universe-spanning meta-organism-highway, made possible by the existence of interstellar or even intergalactic mushroom-like lifeforms connecting to make this “mycelial network.”

The concept appears in “Star Trek: Discovery,” and is, of course, copyrighted by that show’s writers and by the Star Trek franchise itself. But like so many good ideas from Trek, it’s food for thought on which my mind tends to nibble voraciously.

When I thought about my efforts with Ceiling Squares and the problem of figuring out the “Ascent”, or the amount to increase the Ceiling Square of a particular odd integer m in order to find its expression as the difference of two positive integers’ perfect squares, one even and one odd, it occurred to me that I was traveling a vast network of interconnected, interdependent entities, the odd primes and the composite numbers of which they are composed by multiplication, so to speak, and the persistent difficulty mankind has had, so far, in figuring out how much to increase odd positive integer m by trial-adding perfect squares to it until the sum is itself a perfect square (the current equivalent problem to integer factorization, according to my study of this problem).

So of course I, a Trekkie, having only in the past month or so started watching the “Star Trek: Discovery” series, found myself identifying with, or at least aspiring to the genius of, Discovery’s Chief Engineer Paul Stamets (named for a real, living mycologist!), as he and other ship officers discover this network connecting many/most/all of the apparently randomly populated locations of outer space, in order to figure out how to travel from one location to another instantly.

It seemed like a fairly good metaphor, albeit fanciful, as I thought about the unpredictably distributed remainders that appear when one starts “ascending” the Ceiling Square by multiples of 2, in order to find that coveted perfect square remainder: How do I come up with a formula for taking that trip, not by steps of 2, but all at once?

And it’s not just a matter of discovering the Mycelial Network. I’ve seen the perfect squares and remainders, and how they distribute themselves, with the remainders growing according to a simple quadratic as I start increasing the Ceiling Root by 2 and subtracting m from the resulting perfect square. But the Spore Drive is useless as a starship propulsion system if it doesn’t get you where you want to go, and do it faster than you already know how to go.

And today I started thinking about those remainders, and how they are separated by one or more perfect squares, with a few non-square integers before and after all of those perfect-square … jumps …

Yes. The Big Jump I am looking for is a matter of piecing together and understanding all those smaller between-remainder jumps, and then figuring out HOW to figure out the big jump BASED ON where I end up after only the first smaller jump or two. It’s actually a case where knowing exactly where I need to go is the equivalent of getting there. So maybe the unknown destination is more accurately the knowledge of the destination itself.

Metaphors are so fanciful, so exciting! But it’s back to work seeing whether and how they can help me with the real problem at hand. And I don’t have a Giant Space Tardigrade to help navigate.

Stay with me. Let’s get there. In other words, Black Alert.