Oh, I see! So it's the Fourier transform (in the abstract sense) from the integers to the circle group, except that then you extend to all of C (well, possibly

Oh, I see! So it's the Fourier transform (in the abstract sense) from the integers to the circle group, except that then you extend to all of C (well, possibly minus the origin). Interesting! Yeah as someone previously unfamiliar with it, it would have been substantially clearer had you referred explcitly to applying the Z-transform, rather than switching domains, which is the sort of thing that will only make sense to someone who already knows about this. So wait does that mean you weren't actually sure if this approach to that step would work? I assumed you were saying you had a proof, not just outlining an approach you thought would work. (I mean, obviously you didn't have a proof of the whole thing as the overall statement is false, but individual steps might have worked.) But I have to note -- even if the proof works, then if you're applying the Z-transform, then you are once again not sticking to the realm of the discrete! Complex numbers are a continuum matter. So that approach still doesn't yield an integers-only proof! > As to the question about the valid approach, intuitively I see this, but wondering if there was a formal proof of some kind, or is it taken as given? I'm not really sure how to answer this -- what would a formal proof here even consist of? It's easy enough to do it in any instance, but the problem is, how would you even formally state the general principle? Like you could do large classes of statements, certainly. So for instance, if what we're doing is purely algebraic, then you could say, if A and B are algebraic structures, and i:A->B is an injective homomorphism, and S is a set of algebraic equations all of which are always in the image of i, then all of them are always true in A; but of course there's way more types of statements one can make than algebraic equations. So, uh, yeah, one can write down any number of statements like that, but I don't know how you'd formally abstract it into a general principle...?