Notation: fp(n) = period of Fibs mod n. It all boils down to finding fp(p) for p prime, but that's open. I think all we can say is that fp(p) divides p-1 if p=1 or 4 mod 5. fp(p) divides 2(p+1) if p=2 or 3 mod 5. 2(p+1)/fp(p) has to be odd in this last case. (example: if p is a Mersenne prime =2 or 3 mod 5, then 2(p+1) is a power of 2, so fp(p) must equal 2(p+1).)
the highest ratio i've found of period/n is n=10, period=60 the smallest ratio i've found of period/n is n=9349, period=38 (i only generated them up to 10000 for this check)
The max possible value for fp(n)/n is 6 (American Mathematical Monthly Problem E3410, March 1992). See http://www.earlham.edu/~rodrimi/Periods%20of%20Fibonacci%20Sequences%20Mod%2... Equality occurs infinitely often. Gary McGuire