Weakly and Strongly Admissible Triplets for a Collatz-Type Map

TL;DR

This paper extends the classical Collatz problem by introducing weakly and strongly admissible triplets, providing a systematic framework for analyzing convergence and cycles, along with algorithms and conjectures that generalize the original conjecture.

Contribution

It introduces a new class of admissible triplets for Collatz-like problems, establishing structural properties and proposing algorithms and conjectures that extend classical results.

Findings

Bounds on cycle lengths are derived and analyzed.

Algorithms for computing lower bounds on cycle lengths are presented.

Experimental tests support the proposed framework.

Abstract

In this paper, we investigate a class of Collatz-like problems associated with weakly and strongly admissible triplets of integers. This framework extends the classical Collatz mapping, providing a systematic method for generating triplets with convergence to cycles, thereby bypassing the difficulties inherent in solving Diophantine equations. We introduce several special families of admissible triplets and establish general structural properties. In addition, we propose conjectures that generalize the classical Collatz conjecture. Bounds on the lengths of potential non-trivial cycles are derived and analyzed, and two algorithms are presented for computing lower bounds on cycle lengths. Finally, experimental tests are given to illustrate our approach.