Question 158
The Fibonacci sequence is defined by
$$
\left\{\begin{array}{l}
f_{1}=1,\ f_{2}=2 \\
f_{n+1}=f_{n}+f_{n-1},\ n \geq 2
\end{array}\right.
$$
1. Prove that every positive integer can be represented as a sum of several distinct Fibonacci numbers.
2. A positive integer is called Fib-unique if its representation as a sum of several distinct Fibonacci numbers is unique.
Example: $13$ is not Fib-unique because $13=13=8+5=8+3+2$. Find all Fib-unique numbers.
$$
\left\{\begin{array}{l}
f_{1}=1,\ f_{2}=2 \\
f_{n+1}=f_{n}+f_{n-1},\ n \geq 2
\end{array}\right.
$$
1. Prove that every positive integer can be represented as a sum of several distinct Fibonacci numbers.
2. A positive integer is called Fib-unique if its representation as a sum of several distinct Fibonacci numbers is unique.
Example: $13$ is not Fib-unique because $13=13=8+5=8+3+2$. Find all Fib-unique numbers.