Three Dollar Quill

All right, I'm not feeling great today, but I managed to skim that article on algebraic bases, and I'm not sure how much I'll be willing to apply from it. If I'm understanding the article correctly, it establishes by an inductive argument that bases of a particular form have terminating integer representations. However, it appears that, if I base things off of the constructions in the article, then actually obtaining those terminating constructions is going to be linear in the magnitude of the integer. I suppose I could get that to work "better" by caching the conversions, but I'd feel better if I could figure out, say, both incrementing and doubling to construct digit sequences. That way I could ask for, say, one million in phinary, and it would take something like thirty "operations" rather than a million.

So, for sure, it's good to have a restriction on the algebraic bases I want to consider, but I think I'm going to need to do some more research or experimentation. I don't know if this is totally off base (or if I maybe already said something like this before), but the conversion between "base representation and polynomial function of lower degree" feels analogous to the Fourier transform, or inverse Fourier transform, according to my intuition. Just got to... invert an infinite-dimensional matrix? To produce a vector that is sparse in non-zero entries, drawn from a subset of the integers? This idea is not baked all the way.

Anyway, it's late, and I need to sleep.

Good night.