(1) and (2) can be found on the Pisano Period Wikipedia page. The proof for (1) is simple, and (2) is conjectured to be true. (3) is extremely simple since if the period is of length n then that means Fn (mod p) = 0.

With these we can solve the problem by finding all primes such that π(p) divides n and then generating the multiples of all these numbers and checking if their period if equal to n.

A sketch of the algorithm is as follows:

1. Initialize 3 arrays

   1. primes which will store the primes such that π(p) divides n

   2. pivalues which will store π(p)

   3. multiplicities which will store the highest power of p, e, such that π(p^e) divides n 

2. Find the prime factors of Fn 

3. For each prime factor p of Fn  calculate π(p)

   1. If π(p) divides n

      1. Store p in primes

      2. Store π(p) in pivalues

      3. Find maximum e such that π(p^e) divides n and store it in multiplicities 

4. Generate all multiples of the primes stored in primes up to their maximum multiplicities stored in multiplicities together with their corresponding period

5. Go through multiples and sum all numbers with period equal to n

The hardest part was to make an efficient multiples generator. I used recursion, I include the snippet of this function below.