Some results on the Collatz problem

Abstract

The paper refers to the Collatz's conjecture. In the first part, we present some equivalent forms of this conjecture and a slight generalization of a former result from [1]. Then, we present the notion of "chain subtrees" in Collatz's tree followed by a characterization theorem and some subclass of numbers which are labels for some chain subtrees. Next, we define the notion of "fixed points" and using this, we give another conjecture similar to Collatz's conjecture. Some new infinite sets of numbers for which the Collatz's conjecture holds are given. Finally, we present some interesting results related to the number of "even" and "odd" branches in the Collatz's tree.