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The Collatz function as an automorphic Cayley colour graph:decidability of $an+b$ conjectures, proof of the $3n + 1$ conjecture

Jan Kleinnijenhuis, Alissa M. Kleinnijenhuis, Mustafa G. Aydogan

Abstract

The Collatz conjecture states that repeated steps of $n\mathrm{\to }\mathrm{3}n\mathrm{+1}$ at odd numbers and $n\mathrm{\to }n\mathrm{/2}$ at even numbers amount to walks over root paths to the branching number $c=4$ in the `trivial' cyclic root $4\to 2\to 1\to 4\to \dots $ of one connected Collatz graph. The Collatz graph with reverse arrows $n \to 2n$ and $n \to (n-1)/3$ can be transformed to a 3-regular automorphic Cayley color graph $T_{\ge 0}$ with as nodes the branching numbers with a remainder of $4$ or $16$ when divided by $18$, building the congruence classes $[4,16]_{18}$. Labeling the $2^k$ breadth-first ordered root paths with $2^k$ binary numbers on the binary number line, for $k=1,2,3,\dots$, and pairing them with the $2^k$ output numbers of these root paths, gives $2^k$ paired numbers. The 3-regular Cayley graph of these paired branching numbers can be transformed to a 4-regular Middle Pages graph. This 4-regular graph offers to all paired branching numbers from the congruence classes $[4,16]_{18}$ a unique Eulerian tour to and from the trivial root number pair {0,c=4}. This proves Collatz's $3n+1$ conjecture. Whether a specific $an+b$ conjecture offers a Eulerian tour to all its paired branching numbers can be decided by whether it offers such a tour to paired branching numbers lower than $2a^3$.

The Collatz function as an automorphic Cayley colour graph:decidability of $an+b$ conjectures, proof of the $3n + 1$ conjecture

Abstract

The Collatz conjecture states that repeated steps of at odd numbers and at even numbers amount to walks over root paths to the branching number in the `trivial' cyclic root of one connected Collatz graph. The Collatz graph with reverse arrows and can be transformed to a 3-regular automorphic Cayley color graph with as nodes the branching numbers with a remainder of or when divided by , building the congruence classes . Labeling the breadth-first ordered root paths with binary numbers on the binary number line, for , and pairing them with the output numbers of these root paths, gives paired numbers. The 3-regular Cayley graph of these paired branching numbers can be transformed to a 4-regular Middle Pages graph. This 4-regular graph offers to all paired branching numbers from the congruence classes a unique Eulerian tour to and from the trivial root number pair {0,c=4}. This proves Collatz's conjecture. Whether a specific conjecture offers a Eulerian tour to all its paired branching numbers can be decided by whether it offers such a tour to paired branching numbers lower than .

Paper Structure

This paper contains 27 sections, 15 theorems, 26 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Literature review
  3. To be proven
  4. Proof Approach
  5. Picturing composite functions as transformed directed graphs
  6. Elementary number theory
  7. Elementary graph theory
  8. Transforming Collatz graph $G_C$ and 3-regular Cayley graph $T_{\ge 0}$
  9. The planar Cayley graph as an infinitely dimensional fractal binary tree
  10. From $T_{\ge 0}$ to breadth-first tapes of its isomorphic cotrees (Defs.\ref{['eq:leftcotrees']},\ref{['eq:upcotrees']}, Figs.\ref{['fig:ltree']},\ref{['fig:utree']})
  11. From cotrees to breadth-first column tapes in the Middle Pages graph $G_{MP}$
  12. The Collatz congruence classes graph $G_{CC}=([0,\cdots,17]_{18},[f,g])$
  13. Defining the Leftward, Upward and Forward functions
  14. The upward function $U$
  15. The leftward function $L$
  16. ...and 12 more sections

Key Result

Corollary \oldthetheorem

The Syracuse graph $G_{Syr}$ is irregular (Def.def:regular) and a-periodic, even without numbers divisible by $3$ (Def.def:periodens). The Syracuse graph is irregular since odd numbers $o$ do not share the same outdegree, as can be seen in Fig.fig:colga. Depending on whether $o$ is divisible by $3$,

Figures (8)

  • Figure 1: a. The Collatz graph $G_C$ and b. its regular Cayley colour graph $T_{\ge 0}$
  • Figure 2: | Graph transformations yielding the 4-regular middle pages graph $G_{MP}$ (Fig.\ref{['fig:pages']})
  • Figure 3: The Cayley graph $T_{\ge 0}$ as the infinitely dimensional fractal binary tree $T_{CF}$
  • Figure 4: Lefward subtrees $t_{0;\ge i=1,2,3}$ and leftward cotrees $t_{i=1,2,3}$
  • Figure 5: Upward subtrees $T_{\ge i=1,2,3}$ and upward cotrees $T_{i=1,2,3}$
  • ...and 3 more figures

Theorems & Definitions (36)

  • Example \oldthetheorem
  • Definition \oldthetheorem
  • Corollary \oldthetheorem
  • Definition \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • ...and 26 more