Open Problem Garden

Importance: Medium ✭✭

Author(s):

Subject:

Keywords:

Recomm. for undergrads: yes

Prize: $50 (Paid by Clark Kimberling)

Posted	by:	vprusso
	on:	June 12th, 2012

Solved by:

Michael Behrend http://faculty.evansville.edu/ck6/integer/unsolved.html (Problem 17)

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 $u$ .

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 $r$ ?)

Open Problem Garden

Bibliography

Reply

Navigate

Recent Activity