the quadratic extension field , which has p2 elements and whose group of units thus has order p2 − 1, and thus the Pisano period divides p2 − 1. For example, for p = 3 one has π1(3) = 8 which equals 32 − 1 = 8; for p = 7, one has π1(7) = 16, which properly divides 72 − 1 = 48. This analysis fails for p = 2 and p is a divisor of the squarefree part of k2 + 4, since in these cases are zero divisors, so one must be careful in interpreting 1/2 or . For p = 2, is congruent to 1 mod 2 (for k odd), but the Pisano period is not p − 1 = 1, but rather 3 (in fact, this is also 3 for even k). For p divides the squarefree part of k2 + 4, the Pisano period is πk(k2 + 4) = p2 − p = p(p − 1), which does not divide p − 1 or p2 − 1. Fibonacci integer sequences modulo n One can consider Fibonacci integer sequences and take them modulo n, or put differently, consider Fibonacci sequences in the ring Z/nZ. The period is a divisor of π(n). The number of occurrences of 0 per cycle is 0, 1, 2, or 4. If n is not a prime the cycles include those that are multiples of the cycles for the divisors. For example, for n = 10 the extra cycles include those for n = 2 multiplied by 5, and for n = 5 multiplied by 2. Table of the extra cycles: (the original Fibonacci cycles are excluded) (using X and E for ten and eleven, respectively) Number of Fibonacci integer cycles mod n are: 1, 2, 2, 4, 3, 4, 4, 8, 5, 6, 14, 10, 7, 8, 12, 16, 9, 16, 22, 16, 29, 28, 12, 30, 13, 14, 14, 22, 63, 24, 34, 32, 39, 34, 30, 58, 19, 86, 32, 52, 43, 58, 22, 78, 39, 46, 70, 102, ... Notes References External links The Fibonacci sequence modulo m A research for Fibonacci numbers Fibonacci sequence starts with q, r modulo m , a video with Dr. James Grime and the University of Nottingham Fibonacci numbers Modular arithmetic