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.

Sylvain G. writes: I just want to report that I was not able to add the following line with mathjax on my blogger:

Trying this: \begin{align*}\frac{x^n- a^n}{x-a} &= \frac{a^n\left((\frac{x}{a})^n - 1\right)}{a\left(\frac{x}{a}-1\right)}\\ &= a^{n-1}\left( \left(\frac{x}{a}\right)^{n-1} + \left(\frac{x}{a}\right)^{n-2} +\ldots + \frac{x}{a} + 1 \right) \\ &= x^{n-1} + x^{n-2}a + +\ldots + xa^{n-2} + a^{n-1}. \end{align*}

Testing MathJax with TeX Input

Now let's try some displayed math using MathJax (so suitable for most commonly used browsers) with TeX input. \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

That should be it.

Testing MathJax with MathML Input

Now let's try the same displayed math with MathJax enabled (so suitable for most commonly used engines). This time the "math" element has no xmlns attribute. $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

That should be it.

Testing MathML Without MathJax

Now let's try some displayed math without MathJax (so suitable for some Gecko and some Webkit engines). I will use the xmlns attribute, which should be inert with HTML5, in case it helps. $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

That should be it.

Baby Test No. 1

This is a very simple test addressed to the question of how Blogger can be coaxed into accepting raw HTML.

There is no math for now.