Beatty's theoremBeattysTheorem2001-11-06 00:30:112002-05-15 00:49:03Theorem\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}If $p$ and $q$ are positive irrationals such that
$$\frac{1}{p}+\frac{1}{q}=1$$
then the sequences
\begin{eqnarray*}
\{\lfloor np\rfloor\}_{n=1}^\infty&=&\lfloor p\rfloor,\lfloor 2p\rfloor,\lfloor 3p\rfloor,\ldots\\
\{\lfloor nq\rfloor\}_{n=1}^\infty&=&\lfloor q\rfloor,\lfloor 2q\rfloor,\lfloor 3q\rfloor,\ldots\\
\end{eqnarray*}
where $\lfloor x\rfloor$ denotes the floor (or greatest integer function) of $x$, constitute a partition of the set of positive integers.
That is, every positive integer is a member exactly once of one of the two sequences and the two sequences have no common terms.