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

TL;DR

This paper models the Collatz conjecture as an automorphic Cayley color graph, proving the conjecture by demonstrating a unique Eulerian tour for paired branching numbers, and discusses decidability of related $an+b$ conjectures.

Contribution

It introduces a novel graph-theoretic framework transforming the Collatz problem into a Cayley color graph, providing a proof of the $3n+1$ conjecture and a method to decide related conjectures.

Findings

Proves the Collatz $3n+1$ conjecture using graph theory.

Transforms Collatz graphs into Cayley color graphs with Eulerian tours.

Provides a criterion for the decidability of $an+b$ conjectures.

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 $2a^3$.