The thoughts I still get in the early morning hours about numbers and factorization are not so imperative now that I have programmed and published that new, simple algorithm. Sometimes I can just smile at them and go back to sleep. And other times … they’re just WEIRD.
This morning’s stay-awake-and-think-about-it notions concerned the factorizations of even integers, and the strange ways this seems to affect the whole business of that little shortcut of trying to express the integer m as the difference between two perfect squares.
For one strangeness, if m is 2x times an odd integer, and your method chooses that to be the factorization it will use to determine the corresponding difference of perfect squares, you now have to deal with fractions in the answer, which isn’t really proper factorization at all. Or it is, eventually, but the fact that they crop up at all is just . . . A COMPLETELY AVOIDABLE MESS.
Example: m = 12 = 3*4, and that choice of a and b gives an s and t of 7/2 and 1/2, so that m = (72-12)/22. No biggie, you say? It still gives the integer product of 3*4 when we then use the fact that s2-t2=(s-t)(s+t), right?
Fine. Have your fun. It’s just odd to me. It’s much less strangeness in my very strange life to divide 2 completely out to get the m with which I want to start.
Also, what’s with the odd powers of 2?
First of all, 2 to the first power itself has one of those yucky fractional s and t pairs as well: s=3/2 and t=1/2. Same, same, you say? Okay. But now let’s boogie with this.
As soon as the next odd power, 23, something else starts happening:
8 = 32-12.
32 = 62-22.
128 = 122-42.
512 = 242-82.
And so, in general, it looks like, for a positive integer n greater than 1:
22n-1=(3*2n-1)2-(2n-1)2,
which is one of the absolutely weirdest results I’ve gotten from this whole factorization study.
Nah, just factor out The Accursed Dyad, and only deal with odd values of m.
You’ll sleep better. Or maybe I’ll sleep better. Who knows? 😀
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