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Sets forming partition of N iff k is irrational

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October 12, 2010
number theory unsolvednumber theory

Problem Statement

(MOR1)(MOR 1) Denote by [x][x] the greatest integer not exceeding xx. For all real k>1k > 1, define two sequences: an(k)=[nk] and bn(k)=[nkk−1]a_n(k) = [nk]\text{ and } b_n(k) =\left[\frac{nk}{k - 1}\right] If A(k)={an(k):n∈N}A(k) = \{a_n(k) : n \in\mathbb{N}\} and B(k)={bn(k):n∈N}B(k) = \{b_n(k) : n \in \mathbb{N}\}, prove that A(k)A(k) and B(k)B(k) form a partition of N\mathbb{N} if and only if kk is irrational.
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