A Proof of the Collatz Conjecture via Modulo‑4 Congruence Classification

Abstract

The Collatz conjecture, also known as the Hailstone conjecture, states that for any positive integer n, define the iterative transformation rule: divide an even number by 2 directly; if n is odd, calculate 3n+1. All positive integers will converge to 1 after finite iterations.This paper classifies positive integers by modulo‑4 congruence, analyzes the iteration law when odd numbers trigger the formula 3n+1. With the infinite descent method in number theory, we prove iteration values strictly decrease finitely. Restricted by the minimum positive integer 1, iterations converge to 1 ultimately, which completes the proof of Collatz conjecture.