Beatty's theorem
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In mathematics, Beatty's theorem states that if p and q are two positive irrational numbers with
- <math>\frac{1}{p} + \frac{1}{q} = 1,<math>
then the positive integers
- <math>\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<math>
are all pairwise distinct, and each positive integer occurs precisely once in the list. (Here <math>\lfloor x \rfloor<math> 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.