07/20/2022, 10:28 PM
(07/19/2022, 10:03 AM)MphLee Wrote: Idk man, if your having such a strong reaction maybe I don't remember correctly.
But since you're having such a strong reaction maybe it would be interesting to go back to my 2020 notes where I was playing with this idea.
I vaguely remember that it was not so hard to extend \(F_a(n)=a\,{\rm mod}\, n\) from \(n\) integer to \(x\) real. And \(F_a(x)\) had to do with dividing stuff modulo the integers, so in the group of complex roots of units, or something like that. Like the graph of the thing was just a normal hyperbola of the form \(k/x\) but displayed on a quotient of the plane by a 2-lattice, i.e. a torus. In other words it was just division but topologically wrapped up.
Oh yes, I apologize. No, what you are saying here is perfectly possible.
Quote:Maybe the Fields' medal-part of it was that integer points of the hyperbole were important and maybe there is something hard in finding rational/integer points of algebraic curves... it may be that is critical point.
But I don't remember... need to get back to it.
It's the finding the rational/integer solutions that are ridiculously hard. Number theory is all about asking the simplest questions and having 100 pages to almost prove it but only for specific cases, lol.
Ya, solving these iterates in the natural/integer case/describing the equivalence classes. No, that shit is so hard.