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Special M (Solved)

Importance: Medium ✭✭
Author(s): Kimberling, Clark
Subject: Number Theory
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 $?)

Bibliography

http://faculty.evansville.edu/ck6/integer/unsolved.html


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