Primes dividing Fibonacci mod 4

Question

I'm trying to do the following question and am a bit stuck. Would appreciate a hint over a full solution if possible. Here $F_n$ denotes the $n$-th Fibonacci number, and at this point in the question we have already shown $F_{2n}=F_n(F_{n-1}+F_{n+1})$ and $F_{2n+1}=F_n^2+F_{n+1}^2$ for all $n \ge 1$.

Given $m \ge 1$ and $p$ an odd prime dividing $F_m$ determine whether the following statements are true or false:

1. if $m$ is odd, then $p \equiv 1 \pmod 4$

2. if $m$ is even, then $p \equiv 3 \pmod 4$

My line of thinking so far: I know (by Fermat's little theorem) that $x^2 \equiv -1 (\text{ mod p})$ has a solution iff $p \equiv 1 \pmod 4$. So if I can find such an $x$, or show none exist, then I am done. Especially in the $m$ odd case, by writing $m=2n+1$ we have $F_n^2+F_{n+1}^2 \equiv 0 \pmod p$ which implies $(F_n+F_{n+1})^2 \equiv -2F_nF_{n+1} \pmod p$. But I don't see how to progress any further on that front and have a feeling I'm going down a rabbit hole. My other thought is that since p is odd, we know $p$ must be congruent to one of $1$ or $3 \bmod 4$, so we can try and get a contradiction by assuming one or the other. No luck here either.

Answer 1

Your original idea is solid.

The first claim follows from Cassini's identity: $$F_n^2=(-1)^{n+1}+F_{n-1}F_{n+1}$$

To see this, suppose $n+1$ is odd and that $p$ is an odd prime dividing $F_{n+1}$. We instantly get that $$F_n^2\equiv -1\pmod p$$ and we are done.

Answer 2

We have $F_{10}=55$ for $m=10$. Since $p=5$ divides $F_{10}$, but $5\not\equiv 3\bmod 4$, statement $2.$ is false.

Statement 1. is proved in Lemma 2 here - if I understood correctly. Note that it implies that every Fibonacci number $F_{2n+1}$ is the sum of two squares by Fermat's result.