A Unified Proof of the Collatz Conjecture for Positive and Negative Integers

Abstract

The Collatz Conjecture, a long-standing unsolved problem in mathematics, proposes that repeated application of a simple transformation: dividing even numbers by two and mapping odd numbers to three times the number plus one, eventually leads every positive integer to the number one. In this paper, we present a structured, theorem-based approach to proving the conjecture. We first establish that all numbers of the form 2n directly reach 1 through successive divisions by 2. We then prove that every even number reduces to a power of 2 and hence reaches 1. Extending this, we show that odd numbers become even through one Collatz step, allowing the previous results to apply. Finally, we propose a novel extension for negative integers by utilizing 2’s complement mapping, interpreting their trajectories within the framework for positive integers. This comprehensive decomposition offers a unified pathway toward validating the Collatz Conjecture across the full set of integers.