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Observations towards a proof of the Collatz conjecture using $2^{j}k+x$ number series

J. Stöckl

TL;DR

The paper reframes the Collatz conjecture by analyzing sequences with structured OE (odd) and E (even) steps, proving that even starts immediately drop via $n/2$ and deriving rules for odd-start behavior. It introduces a $n=2^{j}k+x$ decomposition, develops a Diophantine framework to compute the first lower number $y$ in a sequence, and classifies cycles by length (notably lengths 6, 8, and a length-16 example) with explicit formulas and subsets. A key result is the derivation that a closed cycle with alternating $O$ and $E$ steps is only possible for the trivial $1\to 4\to 2\to 1$ cycle under certain divisibility constraints, while longer cycles (exemplified by the 27-cycle) illustrate the complexity and potential infinite behavior. The work further argues that the solved-subset probability in $2^{E}+j$ grows toward unity, suggesting a probabilistic route toward showing all starting numbers reach a smaller value and hence ultimately reach 1, though the work remains exploratory and discussion-oriented.

Abstract

The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller number, all full sequences lead to 1 with a finite stopping time, the problem could be reduced to more structured shorter sequences. It is shown that this sequences exist and follow consistent rules. Potential features of an infinite cycle, also leading to a smaller number, are also discussed. Further, an argument for only one possible closed cycle is given for the special sequence of alternating odd and even steps as well as arguments that infinite cycles must exist. Using the observation that periodic behavior is exists an additional argument is provided the probability of subsets which will end up at a number smaller the initial value possibly even of N will indeed end up at unity in the Collatz problem [1]. The work is to be seen as in progress and shared as an contribution to discussion rather than a concrete publication.

Observations towards a proof of the Collatz conjecture using $2^{j}k+x$ number series

TL;DR

The paper reframes the Collatz conjecture by analyzing sequences with structured OE (odd) and E (even) steps, proving that even starts immediately drop via and deriving rules for odd-start behavior. It introduces a decomposition, develops a Diophantine framework to compute the first lower number in a sequence, and classifies cycles by length (notably lengths 6, 8, and a length-16 example) with explicit formulas and subsets. A key result is the derivation that a closed cycle with alternating and steps is only possible for the trivial cycle under certain divisibility constraints, while longer cycles (exemplified by the 27-cycle) illustrate the complexity and potential infinite behavior. The work further argues that the solved-subset probability in grows toward unity, suggesting a probabilistic route toward showing all starting numbers reach a smaller value and hence ultimately reach 1, though the work remains exploratory and discussion-oriented.

Abstract

The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller number, all full sequences lead to 1 with a finite stopping time, the problem could be reduced to more structured shorter sequences. It is shown that this sequences exist and follow consistent rules. Potential features of an infinite cycle, also leading to a smaller number, are also discussed. Further, an argument for only one possible closed cycle is given for the special sequence of alternating odd and even steps as well as arguments that infinite cycles must exist. Using the observation that periodic behavior is exists an additional argument is provided the probability of subsets which will end up at a number smaller the initial value possibly even of N will indeed end up at unity in the Collatz problem [1]. The work is to be seen as in progress and shared as an contribution to discussion rather than a concrete publication.
Paper Structure (19 sections, 30 equations, 6 figures, 16 tables)

This paper contains 19 sections, 30 equations, 6 figures, 16 tables.

Table of Contents

  1. Introduction
  2. Basic Cycles
  3. Basic case: even natural numbers
  4. Basic case: even natural numbers
  5. Sequence Properties
  6. Further cycles
  7. The cycles for the cycle length 6 and first results
  8. The cycles for the cycle length 8
  9. Calculation of the first lower number in a sequence
  10. Subsequent lower number in the same cycle sequence
  11. An example with the length 16
  12. Only one closed cycle
  13. 27 and and more general cycles
  14. The case of 27
  15. The "infinite" sequence
  16. ...and 4 more sections

Figures (6)

  • Figure 1: Basic cycle ending with 1-4-2-1
  • Figure 2: Possible behaviours of Collatz sequences
  • Figure 3: Subsets with sequence (green) and without sequence (blue)
  • Figure 4: Collatz sequences for 575 and 1599
  • Figure 5: Lowest number to fulfill the equation for each $OE$ steps length and respective highest value
  • ...and 1 more figures