I wrote this in an effort to restate some recent basic observations plainly. I am confident that this is no newly-discovered set of mathematical truths, but I am enjoying making these derivations piece by piece and seeing what may follow from each successive one. Perhaps someday I will end up somewhere new. Right now, I am happy just to be browsing in territory Fermat and others have discovered.
Looking at a positive odd integer in two connected ways like this: as the product of two positive integers, and as the difference of two perfect squares, one odd and one even (which comes first depending on the original integer’s value modulo 4), gives me this morning a little thrill because it feels like a primitive ancestor of looking at a sequence of numbers in both the time and frequency domains. It’s a beautiful, simple mathematic duality-in-equivalence.