Proving Coprime nature of Fibonacci numbers I'm trying to prove that Fibonacci numbers $F_i, F_{i+3}$ can be either coprime, where $i$ and $i+3$ are not both even, or they can have a greatest common divisor of $2$, when $i$ and $i+3$ are even.
I've tried using the formula that $gcd(F_m, F_n) = F_{gcd(m,n)}$, but I've not been able to come up with a full proof that ensures that the greatest common factor of Fibonacci numbers 3 terms apart have a maximum greatest common divisor of 2.
Is there any way to prove this maximum of 2, or that $F_i, F_{i+3}$ are coprime, except for when $i$ and $i+3$ are even?

Note that $\gcd(i,i+3) \in \{1,3\}$ because $d \mid i$ and $d \mid i+3$, then $d \mid (i+3)-i=3$. Thus
$$\gcd(F_i, F_{i+3})=F_1 \text{ or } F_3=1 \text{ or } 2.$$
You can also note that $\gcd(i,i+3)=3$ if and only $3 \mid i$.
