The 3n + 1 Problem

A transformation

This is about the transformation (or function) which takes a given positive integer $n$ and produces another as follows:

$$n\mapsto \left\{\begin{array}{cc}3n+1\hfill & \text{if}n\text{is odd}\hfill \\ \frac{n}{2}\hfill & \text{if}n\text{is even}\hfill \end{array}\right.$$ It is sometimes called the Syracuse transformation.

Iteration

One may pick a positive integer as starting point, apply the Syracuse transformation, then apply it again to the result, apply it yet again to the result of that, and so on, to see what eventually happens.

Here is a table of examples:

1 4 2 1 2 1 3 10 5 16 8 4 2 1 5 16 8 4 2 1 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 17 52 26 13 40 20 10 5 16 8 4 2 1 21 64 32 16 8 4 2 1 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1

An open question

Since the 1930s it has been an open question whether for any starting point the sequence generated will eventually reach $1$.

There is a conjecture that every such sequence will eventually reach $1$. It is known by various names, including the “3n + 1 Conjecture”, the “Collatz Conjecture”, the “Syracuse Conjecture”, and others. More information may be found at Wikipedia, https://en.wikipedia.org/wiki/Collatz_conjecture.