[math-fun] Fibonacci mod n question

Consider the Fibonacci sequence F_0 = 0, F_1 = 1, F_2 = 1, ... reduced mod n for fixed n >= 2. This gives a sequence that is clearly periodic, since there are at most n^2 consecutive pairs in Z/nZ. Define fp(n) := the least period of {F_k mod n}. (For example, fp(9) = 24.) QUESTION: What is a formula for fp(n) in general? --Dan Asimov