Question 154
The sets $A$ and $B$ form a partition of positive integers if $A \cap B = \emptyset$ and $A \cup B = \mathbb{N}$. The set $S$ is called prohibited for the partition, if $k + l \neq s$ for any $k, l \in A, s \in S$ and any $k, l \in B, s \in S$.
a) Define Fibonacci numbers $f_{i}$ by letting $f_{1} = 1$, $f_{2} = 2$ and $f_{i+1} = f_{i} + f_{i-1}$, so that $f_{3} = 3$, $f_{4} = 5$ etc. How many partitions for which the set $F$ of all Fibonacci numbers is prohibited are there? (We count $A, B$ and $B, A$ as the same partition.)
b) How many partitions for which the set $P$ of all powers of $2$ is prohibited are there? What if we require in addition that $P \subseteq A$ ?
a) Define Fibonacci numbers $f_{i}$ by letting $f_{1} = 1$, $f_{2} = 2$ and $f_{i+1} = f_{i} + f_{i-1}$, so that $f_{3} = 3$, $f_{4} = 5$ etc. How many partitions for which the set $F$ of all Fibonacci numbers is prohibited are there? (We count $A, B$ and $B, A$ as the same partition.)
b) How many partitions for which the set $P$ of all powers of $2$ is prohibited are there? What if we require in addition that $P \subseteq A$ ?