General Mathematics

    

Cycles in Generalized Collatz Sequences

Authors: Pierre Lamothe

The generalized Collatz sequences are set by D(x) = x/r if x mod r = 0 and T(x) = floor (px/r) otherwise. It has previously been shown (2002.0594) with px + q sequences that numerical cycles are derived from algebraic cycles. The same is shown here in a richer framework where the number of elementary functions increases from 2 to r. Again, the beginning and the end of each sequence are connected by a diophantine equation, pm x - rd y - q = 0, where m and d are the respective numbers of multiplications and divisions. There are still always rotation cycles (q1 q2 ... qm) while derived cycles (x1 x2 ... xm) are present only when qi / (rd - pm) are integers. The function R outlined by Rm(q) = q proves to be a powerful computational tool. In addition, the subsequences are numbered and one can easily find a subsequence from its rank ρ in the base r.

Comments: 26 Pages. En français. 4 figures.

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Submission history

[v1] 2020-06-22 13:34:49
[v2] 2020-06-28 02:05:25

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