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Existence of natural - involves floor function, sqrt{2}

Source: ILL 1979 - Problem 56.

June 5, 2011
functionfloor functionDiophantine equationalgebra unsolvedalgebra

Problem Statement

Show that for every n∈Nn\in\mathbb{N}, n2−⌊n2⌋>12n2n\sqrt{2}-\lfloor n\sqrt{2}\rfloor>\frac{1}{2n \sqrt{2}} and that for every ϵ>0\epsilon >0, there exists an n∈Nn\in\mathbb{N} such that n2−⌊n2⌋<12n2+ϵ n\sqrt{2}-\lfloor n\sqrt{2}\rfloor < \frac{1}{2n \sqrt{2}}+\epsilon.
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