How to tell if a Fibonacci number has an even or odd index Given only $F_n$, that is the $n$th term of the Fibonacci sequence, how can you tell if $n \equiv 1 \mod 2$ or $n \equiv 0 \mod 2$?
I know you can use the Pisano period, however if  $n \equiv 1$ or $n \equiv 2$ $\mod \pi(k)$, it can never be found, where $k$ is in $\pi(k)$ (The Pisano period).
Also there is the fact that if $\sqrt{5F_n^2+ 4}$ is an integer then $n 
\equiv 0 \mod 2$, but is there a faster way?
Lastly, because $F_1 = F_2 = 1$, that would have to be an exception for whatever rule/formula that would apply.

Given some $F_n$, compute: 
1) the residue $f_m = F_n \mod{p}$ where p is a prime such that $x^2-x-1$ has roots in $\mathbb{F}_p$ and $p$ is a safe prime. (for example $p=11$, $p=59$ or $p=179$, ... ).
2) Then calculate the index $m$ of $f_m$ (calculate $f_0 = F_0 \mod{p}, ..., f_{p-1} = F_{p-1} \mod{p}$). 
Result:


*

*If $m$ is even, then $n$ is even, too. 

*If $m=1$, try a different prime.

*else $n$ is odd.

