 Proof of CollatzTheorem

EasyChair Preprint no. 4777

13 pagesDate: December 23, 2020

Abstract

In this article, we will show that Collatz is theorem and we proof it by method that we made in section 2 and 3. In section 1, first we introduction Collatz problem and idea of mathematician about this problem then we change the function of this problem and we make a new definition of Collatz set and generalize Collatz problem in the set theory. In section 2 we decrease all of natural numbers to \$\mathbb{Z}_{10}\$ and make a model with lemma that we said. Then in section 3 we say 3 properties of numbers that are in our models and then we make a new definition of coloring of graph to complete our model and make a new model to explain Collatz system with 3 numbers. Finally in section 4 we begin proof some part of first model and we use properties that we proved in section 3, to proof our model completely.

Keyphrases: Collatz, Collatz Conjecture, collatz system, collatz theorem, Conjecture, even number, finished proof, graph theory, number theory, odd number, output degree, Proof of Collatz, sub-graph