Kimberling, Clark


Special M ★★

Author(s): Kimberling

Let $r$ denote the golden ratio, $\frac{(1+\sqrt{5})}{2}$ and let $\lfloor \rfloor$ denote the floor function. For fixed $n$, let $u(k) = \lfloor kr^n \rfloor$, let $v(k) = \lfloor kr \rfloor^n$, and let $w(k) = \left \lfloor \frac{v(k)}{k^{(n-1)}} \right \rfloor$. We can expect $w$ to have about the same growth rate as $u$.

\begin{conjecture} Prove or disprove that for every fixed $n > 0$, as $k$ ranges through all the positive integers, there is a number $M$ such that $u(k) - w(k)$ takes each of the values $1,2,\dots,M$ infinitely many times, and $u(k) - w(k) \leq M$. (Can you formulate $M$ as a function of $n$? Generalize for other numbers $r$?) \end{conjecture}

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