Beatty’s theorem

If $p$ and $q$ are positive irrationals such that

$$\frac{1}{p}+\frac{1}{q}=1$$

then the sequences

$\mathrm{\{}\lfloor np\rfloor \}{}_{n=1}{}^{\mathrm{\infty }}$ $=$ $\lfloor p\rfloor ,\lfloor 2p\rfloor ,\lfloor 3p\rfloor ,\mathrm{\dots }$

$\mathrm{\{}\lfloor nq\rfloor \}{}_{n=1}{}^{\mathrm{\infty }}$ $=$ $\lfloor q\rfloor ,\lfloor 2q\rfloor ,\lfloor 3q\rfloor ,\mathrm{\dots }$

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.

Title	Beatty’s theorem
Canonical name	BeattysTheorem
Date of creation	2013-03-22 11:56:34
Last modified on	2013-03-22 11:56:34
Owner	drini (3)
Last modified by	drini (3)
Numerical id	6
Author	drini (3)
Entry type	Theorem
Classification	msc 11B83
Related topic	Sequence
Related topic	Irrational
Related topic	Partition
Related topic	Floor
Related topic	Ceiling
Related topic	BeattySequence
Related topic	FraenkelsPartitionTheorem
Related topic	FraenkelsPartitionTheorem2
Related topic	ConjugateIndex