The sequence of Collatz functions, exceptionality of the 3n+1 function and the notion of Collatz generalized Matrix

TL;DR

This paper introduces a sequence of functions similar to the Collatz function, explores their properties, and generalizes the Collatz conjecture to these functions, providing new insights into its behavior and potential proof strategies.

Contribution

It defines a sequence of functions akin to Collatz, studies their properties, and generalizes the Collatz conjecture to these functions, offering new perspectives.

Findings

Collatz conjecture can be extended to a broader class of functions.

Properties of the generalized Collatz functions are characterized.

Behavioral similarities between these functions and the original Collatz function are established.

Abstract

In this article, we define a very important sequence of functions, all the functions of this sequence present behaviors very close to that of the Collatz function. The study of such functions allows us to obtain very interesting results concerning the collatz conjecture and more precisely concerning the generalization of this problem, the nature of the Collatz function, on the statement of the Collatz conjecture we show that this conjecture can be stated for each function on a well-determined subset of Z and other interesting properties in relation to the behaviors of Collatz sequences are studied and proved in this article.