# Canonical decomposition in $\mathbb{r}_+^*$ of a convergent natural number by the collatz iterations

## TL;DR

This paper introduces a canonical decomposition for natural numbers that converge under Collatz iterations, using a function that captures the convergence value for all such sequences.

## Contribution

It constructs a function representing the convergence value and formalizes a canonical decomposition for numbers converging under Collatz iterations.

## Key findings

- A function $f_{X,Y}$ representing convergence values is defined.
- A canonical decomposition for convergent numbers is established.
- The approach provides a new perspective on Collatz convergence patterns.

## Abstract

The Collatz variations pattern seems not to have any recurrence relation between numbers. But knowing that there is at least a natural number that converges after several iterations we construct a function $f_{X,Y}$ that is equal to the value of convergence for all convergent sequences. A canonical decomposition can be expressed for such numbers.