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The Fibonacci numbers are defined by fi = fz = 1 and fn+1 = fn+ fn-1 for n > 2. (a) Prove that god(fn+2,fn) =1 for all n E N. %3D

The Fibonacci numbers are defined by fi = fz = 1 and fn+1 = fn+ fn-1 for n > 2. (a) Prove that god(fn+2,fn) =1 for all n E N. %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The Fibonacci numbers are defined by f1 = f2 = 1 and f(n+1) = fn + f(n-1) for n>2.

The Fibonacci numbers are defined by fi = fz = 1 and fn+1 = fn + fn-1 for n >2. (a) Prove that god(fn+1, Sn) = 1 for all n e N. (b) It is known that the only perfect powers (i.e. p" for some prime p) in the Fibonacci sequence are 1, 8, and 144. Assuming this fact, determine which n are such that fnfn-1 is a square.
Transcribed Image Text:The Fibonacci numbers are defined by fi = fz = 1 and fn+1 = fn + fn-1 for n >2. (a) Prove that god(fn+1, Sn) = 1 for all n e N. (b) It is known that the only perfect powers (i.e. p" for some prime p) in the Fibonacci sequence are 1, 8, and 144. Assuming this fact, determine which n are such that fnfn-1 is a square.
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