TITLE:
Another Proof of the Truth of the Collatz Conjecture, Using Gaussian Arithmetic, Probability, and Statistics
AUTHORS:
Emilio A. Diarte-Carot
KEYWORDS:
Collatz Conjecture, Gaussian Arithmetic, Planar Graphs, Binomial Distribution
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.8,
August
19,
2026
ABSTRACT: In my previous article on the Collatz conjecture, I proved the conjecture without analyzing the structure of the trajectories in detail. It was sufficient to show that they all consisted of movements on a directed graph with six nodes. I used the principles of contradiction and mathematical induction to complete the proof. However, despite the mathematical proof in the aforementioned article, the human imagination can conceive of a realistic trajectory that exhibits phases of growth and decline, but whose overall trend is upward. This paper thoroughly analyzes the structure of the trajectories, using the directed graph mentioned above, a probabilistic decision tree, and statistical concepts, to prove that this trajectory, while appearing realistic, is not feasible. The Collatz conjecture has been proven to be accurate.