 Pisano period

The Pisano period $\pi (m)$ is the period of the Fibonacci number sequence in modular arithmetic modulo $m$, with $F_0=0$ and $F_1=1$. Consecutive residues evolve by the invertible update $(a,b)\mapsto (b,a+b)$ on a finite set, so the pair sequence returns to $(0,1)$ and is periodic from the start.

The Pisano period, denoted as \( \pi(m) \), is the period with which the sequence of Fibonacci numbers repeats modulo \( m \). In other words, if you take the Fibonacci sequence \( F_0, F_1, F_2, \ldots \), and reduce each number modulo \( m \), the resulting sequence will eventually start repeating. The length of this repeating sequence is known as the Pisano period for \( m \).