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 complete file where
Here is the link (to possibly later updated versions) https://go.helms-net.de/math/tetdocs/_ot...ration.pdf , but also uploaded it here.
roots_of_xxx+1_V4_fullexploration.pdf (Size: 1.34 MB / Downloads: 591)
Criticism and constructive ideas are much welcome.
Update 2.9.22 - the method is not yet strong enough to generalize easily to \( \,^4z \pm 1\) and higher; after I found missings in the \( \,^5z + 1 \) case, a sharper look at the assumption of attractivity of the formula 4.1b (in the above linked-to edition) shows an over-generalization of the findings with the \( \,^3z \pm 1 \) case. I'll have to consider now whether -and then how- this problem with attractiveness/contractiveness of the iteration can be fixed at all. So - hold on, you all nice readers... hope this need not be dismissed in whole.
Gottfried
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 complete file where
- I've put together my long answer to the initial problem in MSE (2015, update 2020) pg 1-8
- with the new appendix pg 9-13 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.
Here is the link (to possibly later updated versions) https://go.helms-net.de/math/tetdocs/_ot...ration.pdf , but also uploaded it here.
roots_of_xxx+1_V4_fullexploration.pdf (Size: 1.34 MB / Downloads: 591)
Criticism and constructive ideas are much welcome.
Update 2.9.22 - the method is not yet strong enough to generalize easily to \( \,^4z \pm 1\) and higher; after I found missings in the \( \,^5z + 1 \) case, a sharper look at the assumption of attractivity of the formula 4.1b (in the above linked-to edition) shows an over-generalization of the findings with the \( \,^3z \pm 1 \) case. I'll have to consider now whether -and then how- this problem with attractiveness/contractiveness of the iteration can be fixed at all. So - hold on, you all nice readers... hope this need not be dismissed in whole.
Gottfried
Gottfried Helms, Kassel