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Weakly and Strongly Admissible Triplets for a Collatz-Type Map

Abderrahman Bouhamidi

TL;DR

This work extends the classical Collatz problem by introducing a unified Collatz-type framework built from admissible triplets $(d,\alpha,\beta)_{\pm}$ that define a map $T$. It analyzes weakly and strongly admissible triplets, builds several explicit families with concrete cycle structures, and formulates conjectures that generalize the original problem. The paper derives multiple lower bounds on cycle lengths using Hurwitz-type Diophantine theory, continued fractions, and Farey sequences, and develops algorithms to compute these bounds. A substantial set of computational experiments validates the conjectures across numerous parameter choices and demonstrates the practicality of the lower-bound procedures, signaling a promising path for understanding Collatz-type dynamics beyond the classical case.

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.

Weakly and Strongly Admissible Triplets for a Collatz-Type Map

TL;DR

This work extends the classical Collatz problem by introducing a unified Collatz-type framework built from admissible triplets that define a map . It analyzes weakly and strongly admissible triplets, builds several explicit families with concrete cycle structures, and formulates conjectures that generalize the original problem. The paper derives multiple lower bounds on cycle lengths using Hurwitz-type Diophantine theory, continued fractions, and Farey sequences, and develops algorithms to compute these bounds. A substantial set of computational experiments validates the conjectures across numerous parameter choices and demonstrates the practicality of the lower-bound procedures, signaling a promising path for understanding Collatz-type dynamics beyond the classical case.

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.
Paper Structure (15 sections, 13 theorems, 59 equations, 1 figure, 4 tables, 2 algorithms)

This paper contains 15 sections, 13 theorems, 59 equations, 1 figure, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Weakly and strongly admissible triplets of integers - Definitions
  3. Special classes of admissible triplets and conjectures
  4. The admissible triplets $(d, \alpha,\beta)_{\pmb{\pm}}$ with $\beta =\kappa_0(d^{\nu_0}-\alpha)$, $\alpha=d^{\nu_1}-\kappa_1\delta$, $1\leq \delta\leq d-1$, $\nu_0,\nu_1\geq 1$ and $\kappa_0, \kappa_1=\pm 1$.
  5. The admissible triplets $(d,\alpha,\beta)_{\pmb{+}}$ with $\alpha = d^{\nu_1}+1$ and $\beta=d^{2\mu_0+\nu_1}-\alpha^2$ for $d\geq 2$ with $\mu_0\geq 1$ and $2\mu_0>\nu_1\geq 1$.
  6. The triplets $\bf (d,\alpha,\beta_0)_{\pmb{\pm}}$ and $\bf (d,\alpha,\beta)_{\pmb{\pm}}$ with $\beta= a_0\beta_0$ and $a_0\equiv1\;( {\rm mod}\; d)$.
  7. The special admissible triplets $\bf (d,d+1,-1)_{\pmb{+}}$ and $\bf (d,d+1,1)_{\pmb{-}}$.
  8. The admissible triplets $\bf (d,2d-1,1)_{\pmb{+}}$ evoking the Mersene numbers for $\bf d=2^{p-1}$, $\bf p\geq 2$.
  9. The triplet $\bf (2^p+2^q,2^p+2^{q+1},2^p)_{\pmb{+}}$ with $\bf p\geq q\geq 0$.
  10. Lower bound lengths for cycles associated to an admissible triplet
  11. Experimental tests
  12. First few verifications
  13. Tests for Lower bound length
  14. Verification of the lower bound in Theorem \ref{['Lower_Hurwitz_Bound']}
  15. Conclusion

Key Result

Theorem 2.1

A necessary and sufficient condition that the mapping $T:\mathbb{N}\longrightarrow \mathbb{N}$ given by mapT is well defined from $\mathbb{N}$ into $\mathbb{N}$ is that the triplet $(d,\alpha,\beta)_{\pmb{\pm}}$ associated to $T$ satisfy the following condition

Figures (1)

  • Figure 1: The curves $n\rightarrow \log(R_n(M))$ for different value of $M\in \mathbb{M}$ corresponding to the triplet $(5,6,4)_{\pmb{+}}$.

Theorems & Definitions (37)

  • Conjecture 1.1
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Definition 2.1
  • Conjecture 2.1
  • Conjecture 2.2
  • Theorem 3.1
  • proof
  • Remark 3.1
  • ...and 27 more