Beatty's theorem

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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.)

For example if p=\sqrt 2 and q=2+\sqrt 2 then the sequences are 1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 24, ... (sequence A001951 in OEIS) and 3, 6, 10, 13, 17, 20, 23, 27, 30, 34, 37, 40, 44, 47, 51, 54, 58, ... (sequence A001952 in OEIS).

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.

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