Can Fibonacci Sequences Be Proven Using This Conjecture?

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Discussion Overview

The discussion revolves around the relationship between a specific sequence defined by the recurrence relation A_n = A_{n-1}^{2} - 2 and Fibonacci numbers, specifically whether the product of the sequence can be proven to equal F_{2^{n+1}}. Participants explore potential proofs and references related to this conjecture.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant proposes a conjecture involving the sequence defined by A_1 = 3 and A_n = A_{n-1}^{2} - 2, questioning if the product of the first n terms equals F_{2^{n+1}}.
  • Another participant references a sequence related to the conjecture and requests a proof if it exists.
  • Two identities involving Fibonacci numbers and Lucas numbers are presented as potentially relevant to proving the conjecture.
  • A later reply asks for proof of the identities involving Fibonacci and Lucas numbers.
  • One participant claims that both identities can be proven using closed form expressions for Fibonacci and Lucas numbers, mentioning Binet's formula as a known expression for Fibonacci numbers.

Areas of Agreement / Disagreement

Participants express interest in the conjecture and related identities, but there is no consensus on the existence of a proof or the validity of the conjecture itself.

Contextual Notes

Some assumptions about the relationships between the sequences and the identities may be unverified, and the discussion does not resolve the mathematical steps necessary for a complete proof.

ramsey2879
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Given [tex]A_1 = 3 \texttt{ and } A_n = A_{n-1}^{2} -2[/tex]; is there a way to prove the following:

[tex]\prod_{i=1}^{n}A_{i} = F_{2^{n+1}}[/tex]

or if someone has already proven this, can you give the reference?
 
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Note of is relation can be found at the first comment re the sequence 3,7,47 ... See http://www.research.att.com/~njas/sequences/A001566" , but I would like a proof if it is known.
 
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These two identities should get you there.

[tex] F_{2n}=L_{n}F_{n}[/tex]

[tex] L_{2n}=L_{n}^{2}-2(-1)^{n}[/tex]

Where [tex]L_n[/tex] is the nth Lucas numbers.
 
MrJB said:
These two identities should get you there.

[tex] F_{2n}=L_{n}F_{n}[/tex]

[tex] L_{2n}=L_{n}^{2}-2(-1)^{n}[/tex]

Where [tex]L_n[/tex] is the nth Lucas numbers.
Where can I find a proof of those identities?
 
Both identities can be proven using the closed form expressions for the Fibonacci and Lucas numbers. The closed form expressions should be easily accessible. The one for the Fibonacci numbers is also known as Binet's formula.
 

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