In number theory, the nth Pisano period, written as (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. Definition The Fibonacci numbers are the numbers in the integer sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, ... defined by the recurrence relation For any integer n, the sequence of Fibonacci numbers Fi taken modulo n is periodic. The Pisano period, denoted (n), is the length of the period of this sequence. For example, the sequence of Fibonacci numbers modulo 3 begins: 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, ... This sequence has period 8, so (3) = 8. Properties With the exception of (2) = 3, the Pisano period (n) is always even. A simple proof of this can be given by observing that (n) is equal to the order of the Fibonacci matrix in the general linear group GL2(ℤn) of invertible 2 by 2 matrices in the finite ring ℤn of integers modulo n. Since Q has determinant −1, the determinant of Q(n) is (−1)(n), and since this must equal 1 in ℤn, either n ≤ 2 or (n) is even. If m and n are coprime, then (mn) is the least common multiple of (m) and (n), by the Chinese remainder theorem. For example, (3) = 8 and (4) = 6 imply (12) = 24. Thus the study of Pisano periods may be reduced to that of Pisano periods of prime powers q = pk, for k ≥ 1. If p is prime, (pk) divides pk–1 (p). It is unknown if for every prime p and integer k > 1. Any prime p providing a counterexample would necessarily be a Wall–Sun–Sun prime, and conversely every Wall–Sun–Sun prime p gives a counterexample (set k = 2). So the study of Pisano periods may be further reduced to that of Pisano periods of primes. In this regard, two primes are anomalous. The prime 2 has an odd Pisano period, and the prime 5 has period that is relatively much larger than