In number theory, the nth Pisano period, written as π(n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. The existence of periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774.

The Fibonacci numbers are the numbers in the integer sequence:

defined by the recurrence relation

For any integer n, the sequence of Fibonacci numbers F_i taken modulo n is periodic. The Pisano period, denoted π(n), is the length of the period of this sequence. For example, the sequence of Fibonacci numbers modulo 3 begins:

This sequence has period 8, so π(3) = 8.

With the exception of π(2) = 3, the Pisano period π(n) is always even.

This follows by observing that π(n) is equal to the order of the Fibonacci matrix

in the general linear group ${\displaystyle {\text{GL}}_{2}(\mathbb {Z} _{n})}$of invertible 2 by 2 matrices in the finite ring ${\displaystyle \mathbb {Z} _{n}}$of integers modulo n. Since Q has determinant −1, the determinant of Q^π(n) is (−1)^π(n), which is equal to 1 when either n ≤ 2 or π(n) is even.