Beatty’s theorem

If p and q are positive irrationals such that


then the sequences

{np}n=1 = p,2p,3p,
{nq}n=1 = q,2q,3q,

where x denotes the floor (or greatest integer function) of x, constitute a partitionMathworldPlanetmathPlanetmath 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