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/n. 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:
| n | multiples | other cycles |
|---|---|---|
| 1 | ||
| 2 | 0 | |
| 3 | 0 | |
| 4 | 0, 022 | 033213 |
| 5 | 0 | 1342 |
| 6 | 0, 0224 0442, 033 | |
| 7 | 0 | 02246325 05531452, 03362134 04415643 |
| 8 | 0, 022462, 044, 066426 | 033617 077653, 134732574372, 145167541563 |
| 9 | 0, 0336 0663 | 022461786527 077538213472, 044832573145 055167426854 |
| 10 | 0, 02246 06628 08864 04482, 055, 2684 | 134718976392 |
Read more about this topic: Pisano Period