Question 7
For every integer $k\geq0$, let $F_k$ be the $k$th Fibonacci number, defined by $F_0=0$, $F_1=1$, and $F_k=F_{k-2}+F_{k-1}$ for $k\geq2$. Let $n\geq2$ be an integer, and let $S$ be a set of integers with the following property:
For every integer $k$ with $2\leq k\leq n$, the set $S$ contains two integers $x,y$ such that $x-y=F_k$.
What is the smallest possible number of elements of such a set $S$?
For every integer $k$ with $2\leq k\leq n$, the set $S$ contains two integers $x,y$ such that $x-y=F_k$.
What is the smallest possible number of elements of such a set $S$?