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.