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2013 BAMO12 p5 distinct Fibonacci numbers, D_n = P_{n+1}-P_n,

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August 26, 2019
fibonacci numberSequencealgebra

Problem Statement

Let F1,F2,F3,...F_1,F_2,F_3,... be the Fibonacci sequence, the sequence of positive integers with F1=F2=1F_1 =F_2 =1 and Fn+2=Fn+1+FnF_{n+2}=F_{n+1}+F_n for all n≥1n \ge 1. A Fibonacci number is by definition a number appearing in this sequence. Let P1,P2,P3,...P_1,P_2,P_3,... 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,...1,2,3,4,5,6,8,9,10,13,... since, for example 3=1⋅3,4=2⋅23 = 1 \cdot 3, 4 = 2 \cdot 2, and 10=2⋅510 = 2 \cdot 5. Consider the sequence DnD_n of successive differences of the PnP_n sequence, where Dn=Pn+1−PnD_n = P_{n+1}-P_n for n≥1n \ge 1. The first few terms of D_n are 1,1,1,1,1,2,1,1,3,...1,1,1,1,1,2,1,1,3, ... . Prove that every number in DnD_n is a Fibonacci number.
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