Beatty's theorem

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

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

then the positive integers

$\lfloor p \rfloor, \lfloor 2p \rfloor, \lfloor 3p \rfloor, \lfloor 4p \rfloor, \ldots, \mbox{ and } \lfloor q \rfloor, \lfloor 2q \rfloor, \lfloor 3q \rfloor, \lfloor 4q \rfloor, \ldots$

are all pairwise distinct, and each positive integer occurs precisely once in the list. (Here $\lfloor x \rfloor$ 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.

Categories: Theorems