Taking the fibonacci sequence mod 2 provides this pattern:

Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,…

Sequence mod 2: 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1,…

Taking the fibonacci sequence mod 3 also produces an interesting pattern:

Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,…

Sequence mod 3: 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2,…

Mod 4:

Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,…

Sequence mod 4: 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1,…

This can be continued for mod n.

Consider this table from Pisano Periods – Wikipedia :

(let be the period of fibonacci numbers mod n)

cycle | |||
---|---|---|---|

1 | 1 | 0 | |

2 | 3 | 011 | |

3 | 8 | 0112 0221 | |

4 | 6 | 011231 | |

5 | 20 | 01123 03314 04432 02241 | |

6 | 24 | 011235213415 055431453251 | |

7 | 16 | 01123516 06654261 | |

8 | 12 | 011235 055271 | |

9 | 24 | 011235843718 088764156281 | |

10 | 60 | 011235831459437 077415617853819 099875279651673 033695493257291 |

Note also:

- 2*3=6 which is
- 4*6=24 which is and
- 3*8=24 which is and
- the period for is 20, while the period for is 60. for , and , and ( according to OEIS)

Can we say that ?

Answer: **no.** provides a counter example because and . Therefore, where .

If we extend our table, we can also see that and . In this case, . This also works for , and possibly more. However and do not work. Can we say that ?

Answer: **no,** and do not work. Why are these special cases?

I’ll write a program to test this stuff more when I get home today.