A Collatz Conjecture Proof

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.