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On the stopping time of the Collatz map in $\mathbb{F}_2[x]$

Gil Alon, Angelot Behajaina, Elad Paran

Abstract

We study the stopping time of the Collatz map for a polynomial $f \in \mathbb{F}_2[x]$, and bound it by $O({\rm deg} (f)^{1.5})$, improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence arithmetic sequences of unbounded length in the stopping times of certain sequences of polynomials, a phenomenon observed in the classical Collatz map.

On the stopping time of the Collatz map in $\mathbb{F}_2[x]$

Abstract

We study the stopping time of the Collatz map for a polynomial , and bound it by , improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence arithmetic sequences of unbounded length in the stopping times of certain sequences of polynomials, a phenomenon observed in the classical Collatz map.
Paper Structure (4 sections, 16 theorems, 54 equations)

This paper contains 4 sections, 16 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Bounding the stopping time of the polynomial Collatz map
  3. Arithmetic progressions in stopping times of the Collatz map
  4. Open Questions

Key Result

Theorem 1.1

There exists a constant $c$ such that for all $f \in \mathbb{F}_2[x]$, the stopping time of $f$ with respect to the Collatz map is at most $c\cdot \mathrm{deg}(f)^{1.5}$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Conjecture 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • ...and 23 more