Fibonacci Quadratic Residue

Question

After some research I have came up with a conjecture on Fibonacci quadratic residue:

$F(x)^2 mod F(y)$ =

{ if y is even: $(F(|x-y|)^2$ }

{ if y is odd: $-(F(|x-y|)^2$ }

for values where $|x-y| \le \frac{y+1}{2}$

Where F(n) = the $n^{th}$ Fibonacci number

Questions:

1. Does this apply for all Fibonacci numbers (all y values), and if so, what is the proof?

2. And how can this be extended for values where $|x-y| \gt \frac{y+1}{2}$

I have tested this up to around F10 and the pattern seems to be stable

Examples:

1. For F8:

F8 = 21 so We are looking for numbers $F(n)^2$ mod 21 Here is the sequence starting from n = 0:

(0,1,1,4,9,4,1,1,0,1,1,4,9,4...)
                 A

2. now F9 = 34 so the sequence $F(n)^2$ mod 34 from n=0 is

(0,1,1,4,-25,-9,-4,-1,-1,0,-1,-1,-4,-9,-25,4,1,1,0,1,4,9...) A

Point A marks where x = y