(07/19/2022, 07:59 AM)MphLee Wrote: @James. This times James I don't fully agree since I was able to do something on it... and it seems beautiful... not very hard but the implication it has points to a greater field of study that in my opinion was already fully explored by Grothendieck and algebraic geometers. Keywords should be rational points of algebraic curves, rational functions over the torus and scheme theory.
@Mphlee, well maybe you should get a fields' medal. Let's see that deep Grothendiek construction.
Besides, I meant, there's no analytic iteration--it must be some grothendiek shit. And to fully describe it; you'd definitely get a fields medal... I mean, this question borders the deepest shit within elliptic function theory, and modular function theory. You describe a function that satisfies these kind of modular iterations.... nah bro. You ain't done it.
Restricting to the iterations to \(x \text{mod} y\) and observing \(x,y \in \mathbb{N}\). Where we iterate in \(y\) in some form, is a very deep problem. To describe how it could work is different, than proving it behaves in such a manner. It's part of a program (forget the name) on iteration/natural numbers. It's probably harder than the 3n+1 problem...
Like you can't tell me you've solved:
\[
x\,\text{mod} \left(x \text{mod} p\right) = x \text{mod} y\\
\]
Where \(p = p(x,y) \in \mathbb{N}\) and is effectively a modular square root. It's just not done... Sure it becomes:
\[
x \text{mod} p = y\\
\]
But ffs, even this is a deep problem in number theory. Let alone, if we guess as \(p\to\infty\) in the initial equation. and we talk about prime pairs and prime powers. Trust me, Mphlee. I have a strong background in analytic number theory. If you could answer these questions, the fields medal people would s*** your d***. Lol.