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21

Part of PEN G Problems

Problems(1)

G 21

Source:

5/25/2007
Prove that if α \alpha and β \beta are positive irrational numbers satisfying \frac{1}{\alpha}\plus{}\frac{1}{\beta}\equal{} 1, then the sequences ⌊α⌋,⌊2α⌋,⌊3α⌋,⋯ \lfloor\alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 3\alpha\rfloor,\cdots and ⌊β⌋,⌊2β⌋,⌊3β⌋,⋯ \lfloor\beta\rfloor,\lfloor 2\beta\rfloor,\lfloor 3\beta\rfloor,\cdots together include every positive integer exactly once.
floor functioninequalitiesIrrational numbers