AMM Problem
AMM Problem 11651
Marcel Celaya, Department of Computer Science, University of Victoria.
Frank Ruskey, Department of Computer Science, University of Victoria.
Abstract:
Show that the equality
\[
\left\lfloor \frac{n+1}{\phi} \right\rfloor =
n
- {\small{\left\lfloor \frac{n}{\phi} \right\rfloor}}
+ \biggl\lfloor \frac{ \bigl\lfloor \frac{n}{\phi} \bigr\rfloor}{\phi} \biggr\rfloor
- \biggl\lfloor \frac{ \bigl\lfloor \frac{\left\lfloor \frac{n}{\phi} \right\rfloor}{\phi} \bigr\rfloor}{\phi} \biggr\rfloor
+ \Biggl\lfloor \frac{ \biggl\lfloor \frac{ \bigl\lfloor \frac{\left\lfloor \frac{n}{\phi} \right\rfloor}{\phi} \bigr\rfloor}{\phi} \biggr\rfloor}{\phi} \Biggr\rfloor
- \cdots
\]
holds for all non-negative integers $n$
if and only if $\phi = (1+\sqrt{5})/2$, the golden ratio.
Comments:
-
Appears as problem 11651,
in the June-July 2012 issue of the American Mathematical
Monthly, on page 522.
Selected papers that refer to this problem:
-
Someday there might be some!