The Pisano period of a prime , is defined as the minimum integer satisfying , where is the Fibonacci sequence. It is well known that for , is either a divisor of or . In this article, we extend this conclusion to any C-finite sequence of the formulae , where is a constant.
Closed form expression
Assume has a general form . Then we rewrite the recursive formulae as , which yields a characteristic equation . Let be the characteristic roots. Assume , W.L.O.G., We claim that:
Proof: since and ,
Pisano period
As aforementioned, the characteristic roots should satisfy or . If such that , it is straightforward to derive and and thus . Then by Fermat’s little theorem, and:
which implies that the Pisano period .
If such that , we claim that .
Proof: since are the roots of the equation , and .
Let . Then:
Note that and . Therefore, by Fermeat’s little theorem: