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Paradoxical behavior in Collatz sequences

Olivier Rozier, Claude Terracol

TL;DR

The paper investigates paradoxical behavior in Collatz sequences by analyzing the linear relation $T^{j}(n)=C_j(n)\,n+E_j(n)$ with $C_j(n)=\frac{3^{q}}{2^{j}}$ and the remainder $E_j(n)$. It introduces a parity-vector framework and an unordered-majorization-based partial order to relate the remainder $E_j(n)$ to parity patterns, linking paradoxical growth to near-equalities of $3^{q}$ and $2^{j}$ via continued fractions of $\frac{\log 2}{\log 3}$. Computational results identify 593 paradoxical sequences for $7\le n\le 4614$ and show no such sequences up to much larger bounds, while several conditional results indicate that finiteness of paradoxical sequences with first term $>2$ would imply the Collatz conjecture. A heuristic analysis, supported by delay/maximum-excursion bounds and Rhin’s lower bounds on linear forms in logarithms, argues that paradoxical sequences are finitely many and likely do not exist beyond $n=4614$, suggesting a stronger statement than the classical Collatz or Terras’ CST. These findings illuminate the deep connections between CST, parity-pattern dynamics, and global convergence in Collatz-type maps.

Abstract

On the set of positive integers, we consider an iterated process that sends $n$ to $\frac{3n+1}{2}$ or to $\frac{n}{2}$ depending on the parity of $n$. According to a conjecture due to Collatz, all such sequences end up in the cycle $(1,2)$. In a seminal paper, Terras further conjectured that the proportion of odd terms encountered when starting from $n\geq2$ is sufficient to determine its stopping time, namely, the number of iterations needed to descend below $n$. However, when iterating beyond the stopping time, there exist "paradoxical" sequences for which the first term is unexpectedly exceeded. In the present study, we show that this topic is strongly linked to the Collatz conjecture. Moreover, this non-typical behavior seems to occur finitely many times apart from the trivial cycle, thus lending support to Terras' conjecture.

Paradoxical behavior in Collatz sequences

TL;DR

The paper investigates paradoxical behavior in Collatz sequences by analyzing the linear relation with and the remainder . It introduces a parity-vector framework and an unordered-majorization-based partial order to relate the remainder to parity patterns, linking paradoxical growth to near-equalities of and via continued fractions of . Computational results identify 593 paradoxical sequences for and show no such sequences up to much larger bounds, while several conditional results indicate that finiteness of paradoxical sequences with first term would imply the Collatz conjecture. A heuristic analysis, supported by delay/maximum-excursion bounds and Rhin’s lower bounds on linear forms in logarithms, argues that paradoxical sequences are finitely many and likely do not exist beyond , suggesting a stronger statement than the classical Collatz or Terras’ CST. These findings illuminate the deep connections between CST, parity-pattern dynamics, and global convergence in Collatz-type maps.

Abstract

On the set of positive integers, we consider an iterated process that sends to or to depending on the parity of . According to a conjecture due to Collatz, all such sequences end up in the cycle . In a seminal paper, Terras further conjectured that the proportion of odd terms encountered when starting from is sufficient to determine its stopping time, namely, the number of iterations needed to descend below . However, when iterating beyond the stopping time, there exist "paradoxical" sequences for which the first term is unexpectedly exceeded. In the present study, we show that this topic is strongly linked to the Collatz conjecture. Moreover, this non-typical behavior seems to occur finitely many times apart from the trivial cycle, thus lending support to Terras' conjecture.

Paper Structure

This paper contains 6 sections, 13 theorems, 61 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. A partial order on parity vectors
  3. From divergent to paradoxical sequences
  4. Properties of paradoxical sequences
  5. Computational results
  6. Heuristic analysis

Key Result

Theorem 1.1

There are exactly 593 paradoxical sequences starting with an integer $n$ such that $7 \leq n \leq 4614$ and there is no other with $3 \leq n \leq 2.8 \times 10^{19}$. Moreover, if the number of paradoxical sequences with first term greater than 2 is finite, then the Collatz conjecture is true.

Figures (1)

  • Figure 1: Hasse diagrams of (a) the total order on parity vectors of length 3 and ones-ratio $\frac{1}{3}$, and of (b-d) the partial orders on parity vectors of length 4, 5, 6 and ones-ratios $\frac{1}{2}$, $\frac{3}{5}$, $\frac{1}{2}$, in that order.

Theorems & Definitions (35)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • ...and 25 more