Proof of the Collatz Conjecture
Abstract
The Collatz conjecture (3n+1) has been one of the most famous unsolved problems in mathematics since it was proposed in 1937. In this paper, we present a complete proof of this conjecture using a new representation of odd numbers in the form n = 2k+1. We uncover a fundamental alternation property in the behavior of these numbers, and show that every odd number inevitably leads to a smaller number within at most two or three steps. Using this property, we prove that all natural numbers eventually reach the cycle 4-2-1.
Fields of Study & Subjects
Smarandache Functions and Recreational Number Theory
Analytic Number Theory Research
Mathematical Dynamics and Fractals
Publication details
- Journal / source
- Zenodo (CERN European Organization for Nuclear Research)
- Publication date
- 26 February 2026
- Language
- EN
- Document type
- Preprint