Mathematical Insights, an Update: Ceiling Squares and Central Squares


But First: A Very Personal Psychological/Spiritual/Mystical/I-Really-Don’t-Know-What-Exactly-It-Is Side Trip

My mathematical insights are still coming fairly predictably almost every night, trains of thought that feel inspired and that push me to think them to completion, or at least to some kind of form. My Primary Care Physician sees my state as being similar to that of certain sects of the Sufi, who achieve a kind of mystical ecstasy. I do not know that mathematical inspirations count as mystical, but the mood and the experience I have had of dealing with all of this, yes, it does feel mystical, or perhaps a bit like a near-enlightenment experience.

In any case, it is not the Holy Spirit driving me, I don’t think, or some loa from Haitian voodoo riding me as their “horse.” No, if there is any personification, it might be the Queen of the Sciences Herself visiting me, or perhaps the Angels that Rebecca Newberger Goldstein’s fictional character Azarya imagined the prime numbers to be. He was the son of a Rabbi of an again fictional Ultra-Orthodox Jewish sect in the additionally fictional community of New Walden, in the Hudson Valley of New York. Just to show you how weird coincidences factor into all of this as well, I met my wife Wendy when she lived in the very real town of Walden (not New Walden), and her grandchildren, who have watched my recent mathematical studies on my MacBook Pro with great interest albeit not-great understanding, live with their mother, Wendy’s daughter, and their father in a lovely home in New York State’s lovely Hudson Valley.

But anyway, this whatever-it-is possessing me, or driving me, in these mathematical inspirations and the objects, constructs, theories, visualizations that result? Maybe it’s related to sleep deprivation more than anything else. Who knows? Doc says I’m mostly all right. His prescriptions are really suggestions for me to consider. He’s a good Primary Care Doctor. And I am feeling really good in so many other ways. I do think it’s something like a near-enlightenment experience, or that Dervish spiritual ecstasy. But now, I think I’ve reached the end of what I wanted to explain on that tangent :-D, so let’s do some math.

And Now, the Mathematics

Consider an odd integer m, along with a way of factoring it into two other odd integers a and b, with a<b. The number of ways to do this number roughly half of the divisors of m, to include 1 and m itself.

I have defined elsewhere and talked about the ceiling square of m, which is the least perfect square greater than m. The ceiling square can be odd or even, and when one subtracts m from it, one obtains a remainder which is even or odd with respect to the ceiling square.

As I have also explained elsewhere, all of this is much neater and more consistent when one deals with odd integers only. The integer factors of an odd integer must themselves be odd, to start with. And when I was looking last week at the telescoping of m into a kind of nested difference of perfect squares, the series of ceiling squares that made this possible seemed to hit an odd condition, whenever a remainder came about that was exactly 2, where at that point the series of ceiling squares would telescope infinitely: a succession of remainders of 2 that never ended, relating to the observation that 2 is the remainder when you subtract 2 from its ceiling square 4. (Don’t worry, only the solutions of x2-2x=0 can do that, they are 0 and 2, and 0 is a perfect square and thus a stopping point in the series.)

This week, my mathematical inspiration changed its focus so that I do not have to deal with The Accursed Dyad (as wicked old Aleister Crowley generally described the number 2) any more, at least for now. Along with m’s ceiling square, I have begun thinking about the central square associated with the factorization m=ab, and given by the quantity s=(a+b)/2. Since a and b, as I said, are odd, this is always an integer, and it is an integer for which we can always find a representation of m as the difference s2-t2 for some odd integer t<s.

Furthermore, each different factorization of m into a times b will yield a different s and t, and there will be a one to one correspondence between them, so that each such possible s2-t2 corresponds with one and only one set of factors a and b.

I think that’s neat, but I’m not fooling myself: These numbers and their behavior correspond with nothing more fancy, really, than the numerical behavior of the Sieve of Eratosthenes. A day or three ago, I constructed a visualization of this and had the notion to call it the Tree of Eratosthenes due to the way parts of the visualization seemed to branch out. But, as I said, it has not yet led to any profound revelation or deep insight beyond the simplest of implication of the divisors of m and how they yield different pairs of ab=m.

Oh, well, if more thoughts come – I feel certain this math craziness has a good chance of continuing, and I’ll be here to be “ridden by it” – I will assemble a Google Doc paper to report on it.

Stay tuned. This horse ain’t headin’ for the stable yet.

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