(08/28/2022, 12:00 PM)Gottfried Wrote: Update for the initial posting.
Just played around with this problem (main question in MSE: https://math.stackexchange.com/questions...olve-xxx-1 ) again, and by a sudden idea - reflecting my "invention" of the iterated-branched-logarithm in the "periodic-points" thread - I got now apparently the complete solution for the finding of roots in \( \,^3z +1 \), and the generalizations for the finding of roots in \( \,^3z \pm 1 \) and \( \,^4z \pm 1 \); the scheme is so simple that its further generalization to \( \,^mz \pm 1 \) is simply an extension of a couple of lines of code.
I've not yet time to set this up in mathjaxform, to make this visible directly here in the forum-box, but I upload
- the basic file where I've put together my long answer to the initial problem in MSE (2015, update 2020) pg 1-8
- the appendix pg 9-12 only containing my solution which I found yesterday & today. This is heuristic, and perhaps spurious cases missing - don't know how to handle the problem to prove to really have the exhaustion of all the possible roots.
Criticism and constructive ideas are much welcome.
Gottfried
Just flipping through this initially. This is super cool. The day we start relating "roots of tetration" and "Number theory" we can finally get some traction. The way we are just flippantly talking about \(^m z \pm 1\)--it's going to be so valuable. Not much else to add, but dope as hell, Gottfried
Imagine we get an Euler's theorem--no one ever (way back in the day) guessed that \(z^m \pm 1=0\) would ever just become \(z^m + 1=0\) -- thru a "change of variables"
