A Collatz Conjecture Proof

TL;DR

This paper introduces a novel recursive representation of the Collatz function, constructs a complete Collatz tree, and provides a proof of the Collatz conjecture using difference equations and the axiom of choice.

Contribution

It presents a new recursive formulation, constructs a complete Collatz tree, and offers a proof of the Collatz conjecture.

Findings

Complete Collatz tree constructed for odds not divisible by 3

Proof of the Collatz conjecture provided

Reduced subsequences contain fewer elements than original sequences

Abstract

We represent the generalized Collatz function with the recursive ruler function r(2n) = r(n) + 1 and r(2n + 1) = 1. We generate even-only and odd-only Collatz subsequences that contain significantly fewer elements term by term, to 2 and 1, respectively, than are present in the original 3n + 1 and the Terras-modified Collatz sequences. We show that a nonlinear, coupled system of difference equations yields a complete acyclic (except for the trivial cycle) Collatz tree in odds not divisible by 3 with root vertex 1. We construct a complete Collatz tree with the axiom of choice and prove the Collatz conjecture.