Introduction

Preliminaries on the Collatz Conjecture.

Collatz function $F : \mathbb{N} \to \mathbb{N}$ defined as:

$$F(n) = \begin{cases} 3n+1 & n \equiv 1 \bmod 2 \\ \frac{n}{2} & n \equiv 0 \bmod 2\end{cases}$$

The problem asks whether $\exists j \in \mathbb{N},\forall n \in \mathbb{N} : F^j(n) = 1$. We still don't know if this is true (since 1937), but it is conjectured to be so by many. In this work, we will approach the problem in two ways:

• Using binary trees to hope for a proof-by-contradiction opportunity

• Extending the operation to real numbers and analyzing patterns

Reducing to Odd Numbers

It is immediate to observe that showing the conjecture of odd numbers suffices; even numbers eventually reduce to odd numbers via $n/2$. For this purpose, we have the following two functions:

• Reduced Collatz function $R : \mathbb{O} \to \mathbb{O}$ defined as:

$$R(n) = \frac{3n+1}{2^{x}}$$

where $x$ is the largest number such that result is an odd number.

• Reduced Collatz Initializer function $R_0 : \mathbb{N} \to \mathbb{O}$ defined as:

$$R_0(n) = \frac{n}{2^x}$$

where $x$ is the largest number such that result is an odd number.