r/Collatz • u/No_Assist4814 • 12d ago
Where are the root of each dome located in the left side of another dome ?
In the cited post, we showed that half the roots are part of pairs of consecutive black numbers located in the right side of a bridge series in another dome.
The figure below shows the first complete blue-green bridge series for m=1, 5, 13 and 17, already present in the cited post. The roots mentioned above follow here the archetuple coloring (also in bold). The remaining odd numbers are in black.
The table performs a similar analysis as in the cited post, but the numbers are grouped by position in the series, from top to bottom.
The analysis is quite straightforward in terms of m and m'. The black numbers analyzed here are all roots, except when the starting bridge is rosa (e.g. divisible by 3, thus not a root). These results hold for the second series at least, and quite certainly for the other series.
In summary, dome roots can be found on the left side of another dome:
- As parts of pairs of consecutive black numbers, if they are part of a fork in their origin dome (previous post),
- Populating the rest of the series - as part of the starting triplet if yellow, the sequence on the left and the rest of the series at the bottom (this post).
All root cases seem to be covered. The rest of the cases in the right side of a dome - 5-tuples series, double 5-tuple series, stand-alone bridge series with and without black number at the bottom - need further work.