Beatty's theorem

In mathematics, '''Beatty's theorem''' states that if ''p'' and ''q'' are two negative and non-negative numbers 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. Theorems

Beatty's theorem

== Notation == I am not very familiar with math notations. What doe

1p

mean? 1*p or 1 prefixed before p? -- User:Sundar 10:50, Dec 8, 2004 (UTC) :1 times ''p''. Or, in other words, simply ''p''. It's kind of redundant to specify multiplication by one, but it was probably done to clarify the sequence. User:Gwalla — User:Gwalla | User talk:Gwalla 17:13, 6 Jan 2005 (UTC) ::IMO that sequences are clear enough even without it, so I'm going to delete it. == Diagram for Beatty's theorem == Here's a pair of diagrams which neatly illustrate the theorem. www3.telus.net/ldh/math/beatty.htm Maybe somebody who knows Wiki better than I do can insert these diagrams, or similar ones, into the article. Notice also that the theorem can be extended from lines of the form y/x = positive irrational constant to curves that go through no lattice point.

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