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Recurrence relation, Fibonacci numbers

could someone possibly help me prove. thankyou.

$(a)$ Consider the recurrence relation $a_{n+2}a_n = a^2 _{n+1} + 2$ with $a_1 = a_2 = 1$.

prove $a_n$ and $a_{n+1}$ are coprime for $n \in \mathbb N$

so far i have:

$a_1 = a_2 = 1$

$a_3 = 3$

$a_4 = 11$

$a_5 = 41$

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    james, see also http://math.stackexchange.com/questions/240724/fibonacci-question?lq=12012-11-20

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For every $n$, $a_{n+2}\color{red}{a_n}-a_{n+1}\color{red}{a_{n+1}}=2$ hence Bézout says that the gcd of $\color{red}{a_n}$ and $\color{red}{a_{n+1}}$ is either $1$ or $2$. Since every $a_n$ is odd, this gcd is $1$.

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    @Bill: No, I never had a name for it at all.2012-11-21