Collatz Conjecture

Conjecture

 * Collatz conjecture.png

Let $f: \N \to \N$ be the mapping defined on the natural numbers as follows:


 * $\forall n \in \N: f \left({n}\right) = \begin{cases}

n / 2 & : n \text { even} \\ 3 n + 1 & : n \text { odd} \end{cases}$

For any given value of $n$, let the sequence $\left\langle{S_k}\right\rangle$ be defined:


 * $\forall k \in \N: S_k = \begin{cases}

n & : k = 0 \\ f \left({S_{k-1}}\right) & : k > 0 \end{cases}$

The Collatz conjecture is:


 * For all $n \in N$, there exists some $i \in \N$ such that sequence $S_i = 1$.

That is, by repeatedly applying the rule: half it if even, otherwise multiply it by $3$ and add $1$ to any natural number, eventually you will reach $1$.

Progress
As of $7$th March $2017$, all numbers up to $2^{60}$ have been tested, and all end in $1$.