# 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

Note of is relation can be found at the first comment re the sequence 3,7,47 .... See A001566, but I would like a proof if it is known.

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.