Are long series of series of preliminary pairs possible ? II
Follow up to [Are long series of series of preliminary pairs possible ? : r/Collatz](https://www.reddit.com/r/Collatz/comments/1ks6u5m/are_long_series_of_series_of_preliminary_pairs/)
In that post, I hypothetized that there were two cases: isolated longer series and joined series of shorter series, based on many examples I have.
To be on the safe side, I looked for counter-examples and I did not have to go very find to find some.
The figure below shows shorter series (3/5 numbers) and longer series (5/7 numbers) being joined (left). The black cell indicates the triangle the series belong to.
Keep in mind that this is a partial tree. Notably, the even numbers in the colored partial sequences iterate also from their double and so on on their right. But the odd numbers in the left colored partial sequences cannot form tuples on their left.
Note the special case of 50 that belongs to two series, perhaps as it is part of an odd triplet.
The same partial tree in mod 48 allows to identify the segments involved.
In summary, series of series involving longer series are possible.
[Overview of the project (structured presentation of the posts with comments) : r/Collatz](https://www.reddit.com/r/Collatz/comments/1k98c5a/overview_of_the_project_structured_presentation/)
Posted May 22, 2025, 9:27 AM UTC