Beatty's theorem

Beatty's theorem states that if p and q are two positive irrational numbers with

then the positive integers

are all pairwise distinct, and each positive integer occurs precisely once in the list. (Here denotes the floor function of x, the largest integer not bigger than x.)

The theorem was published by Sam Beatty in 1926.

The converse of the theorem is also true: if p and q are two real numbers such that every positive integer occurs precisely once in the above list, then p and q are irrational and the sum of their reciprocals is 1.