The Proof of the Collatz Conjecture

Abstract

We examine the Collatz sequence by expressing any positive integer as 𝑛 = 2𝑘 × 𝑚, with 𝑚 odd and 𝑘 ≥ 0, thereby unveiling a recurring structure. When 𝑛 is a pure power of two (i.e., 𝑚 = 1), the sequence simplifies dramatically: repeated halving deterministically leads to 2 0 (which equals 1). We extend this observation by representing each transformation—whether halving an even number or applying the 3𝑛 + 1 rule to an odd number—as a directed edge in a graph where each node corresponds to an integer and each branch signifies a unique pathway from the sequence’s origin to its eventual convergence at 1. Notably, although some even numbers admit two predecessors, every integer has exactly one successor, which naturally organizes the graph as an inverted tree with its root at 1. This graph-theoretical framework not only elucidates the well-behaved nature of powers of two, but also provides a systematic method for analyzing the more complex transitions of odd numbers. Ultimately, we prove that the root 1 can reach all positive integers, and all positive integers can reach the root 1 on the inverted tree, thereby establishing the Collatz conjecture

Keywords: Collatz conjecture, 3𝑛 + 1 problem, Tree structures, Discrete dynamical systems, Convergence analysis