Sharkovskii’s theorem
Every natural number^{} can be written as ${2}^{r}p$, where $p$ is odd, and $r$ is the maximum exponent^{} such that ${2}^{r}$ divides (http://planetmath.org/Divisibility) the given number. We define the Sharkovskii ordering of the natural numbers in this way: given two odd numbers^{} $p$ and $q$, and two nonnegative integers $r$ and $s$, then ${2}^{r}p\succ {2}^{s}q$ if

1.
$$ and $p>1$;

2.
$r=s$ and $$;

3.
$r>s$ and $p=q=1$.
This defines a linear ordering of $\mathbb{N}$, in which we first have $3,5,7,\mathrm{\dots}$, followed by $2\cdot 3$, $2\cdot 5,\mathrm{\dots}$, followed by ${2}^{2}\cdot 3$, ${2}^{2}\cdot 5,\mathrm{\dots}$, and so on, and finally ${2}^{n+1},{2}^{n},\mathrm{\dots},2,1$. So it looks like this:
$$3\succ 5\succ \mathrm{\cdots}\succ 3\cdot 2\succ 5\cdot 2\succ \mathrm{\cdots}\succ 3\cdot {2}^{n}\succ 5\cdot {2}^{n}\succ \mathrm{\cdots}\succ {2}^{2}\succ 2\succ 1.$$ 
Sharkovskii’s theorem. Let $I\subset \mathbb{R}$ be an interval, and let $f:I\to \mathbb{R}$ be a continuous function^{}. If $f$ has a periodic point^{} of least period $n$, then $f$ has a periodic point of least period $k$, for each $k$ such that $n\succ k$.
Title  Sharkovskii’s theorem 

Canonical name  SharkovskiisTheorem 
Date of creation  20130322 13:16:11 
Last modified on  20130322 13:16:11 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  7 
Author  Koro (127) 
Entry type  Definition 
Classification  msc 37E05 
Synonym  Sharkovsky’s theorem 
Defines  Sharkovskii’s ordering 
Defines  Sharkovsky’s theorem 