This week I did some tests regarding the Pisano period (and a question at MSE) and I found the following four conditions to be true for the first 10.000 values of pi(n) (Pisano period for n).
I do not understand the reasons for the results: points 1-3 would work as a primality test, but it does not detect all the possible primes, only a subset of them, e.g. {2,5,47,107,113,139,…}do not comply with points 1-3 and are not detected. And specially the last point, if the test is correct, would mean that the Pisano period of a Fibonacci prime is exactly four times the index of the Fibonacci prime in the Fibonacci sequence when the index is greater than 5 (being F5=5) . For instance: pi(1597)=68 and 68/4=17 which is exactly the index of 1597 in the Fibonacci sequence, F(17)=1597.
According to the answer and comments in the question at MSE it is not clear if it would be true or not yet (there could be big pseudoprime counterexamples). After doing the question, I noticed that the first one (1) was already registered at the OEIS sequence in the "formula" section. It does not mean that it is correct, but at least nobody removed that information since 2002, so it might be significative. My other observations are not registered at OEIS though.
According to the answer and comments in the question at MSE it is not clear if it would be true or not yet (there could be big pseudoprime counterexamples). After doing the question, I noticed that the first one (1) was already registered at the OEIS sequence in the "formula" section. It does not mean that it is correct, but at least nobody removed that information since 2002, so it might be significative. My other observations are not registered at OEIS though.
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