Question 23
Let $F_{1}, F_{2}, F_{3} \ldots$ be the Fibonacci sequence, the sequence of positive integers with $F_{1}=F_{2}=1$ and $F_{n+2}=F_{n+1}+F_{n}$ for all $n \geq 1$. A Fibonacci number is by definition a number appearing in this sequence.
Let $P_{1}, P_{2}, P_{3}, \ldots$ be the sequence consisting of all the integers that are products of two Fibonacci numbers (not necessarily distinct), in increasing order. The first few terms are
$$
1,2,3,4,5,6,8,9,10,13, \ldots
$$
since, for example $3=1 \cdot 3, 4=2 \cdot 2$, and $10=2 \cdot 5$.
Consider the sequence $D_{n}$ of successive differences of the $P_{n}$ sequence, where $D_{n}=P_{n+1}-P_{n}$ for $n \geq 1$. The first few terms of $D_{n}$ are
$$
1,1,1,1,1,2,1,1,3, \ldots
$$
Prove that every number in $D_{n}$ is a Fibonacci number.
Let $P_{1}, P_{2}, P_{3}, \ldots$ be the sequence consisting of all the integers that are products of two Fibonacci numbers (not necessarily distinct), in increasing order. The first few terms are
$$
1,2,3,4,5,6,8,9,10,13, \ldots
$$
since, for example $3=1 \cdot 3, 4=2 \cdot 2$, and $10=2 \cdot 5$.
Consider the sequence $D_{n}$ of successive differences of the $P_{n}$ sequence, where $D_{n}=P_{n+1}-P_{n}$ for $n \geq 1$. The first few terms of $D_{n}$ are
$$
1,1,1,1,1,2,1,1,3, \ldots
$$
Prove that every number in $D_{n}$ is a Fibonacci number.