# Beatty’s theorem

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

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

then the sequences

 $\displaystyle\{\lfloor np\rfloor\}_{n=1}^{\infty}$ $\displaystyle=$ $\displaystyle\lfloor p\rfloor,\lfloor 2p\rfloor,\lfloor 3p\rfloor,\ldots$ $\displaystyle\{\lfloor nq\rfloor\}_{n=1}^{\infty}$ $\displaystyle=$ $\displaystyle\lfloor q\rfloor,\lfloor 2q\rfloor,\lfloor 3q\rfloor,\ldots$

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