Conjecture re Fibonacci numbers

1. Apr 22, 2008

ramsey2879

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

$$\prod_{i=1}^{n}A_{i} = F_{2^{n+1}}$$

or if someone has already proven this, can you give the reference?

2. Apr 23, 2008

ramsey2879

Last edited by a moderator: Apr 23, 2017
3. Apr 23, 2008

MrJB

These two identities should get you there.

$$F_{2n}=L_{n}F_{n}$$

$$L_{2n}=L_{n}^{2}-2(-1)^{n}$$

Where $$L_n$$ is the nth Lucas numbers.

4. May 15, 2008

ramsey2879

Where can I find a proof of those identities?

5. May 17, 2008

MrJB

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.