The Pisano period is the period of the Fibonacci number sequence in modular arithmetic modulo , with and . Consecutive residues evolve by the invertible update on a finite set, so the pair sequence returns to and is periodic from the start.
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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 \( m \).