### Abstract

Given a prime p ≥, and given 1 < κ < p-1, we call a sequence (a_{n})_{n} in F_{p} a Φκ-sequence if it is periodic with period p-1, and if it satisfies the linear recurrence a _{n} + a_{n}+1 - a_{n+κ} with a_{0} = 1. Such a sequence is said to be a complete Φκ-sequence if in addition {a_{0},a_{1}, ⋯, a_{p-2}} = {1, ⋯,p-1}. For instance, every primitive root b mod p generates a complete Φ_{κ}-sequence a_{n} = b^{n} for some (unique) κ. A natural question is whether every complete Φ_{κ}- sequence is necessarily defined by a primitive root. For κ = 2 the answer is known to be positive. In this paper we reexamine that case and investigate the case κ = 3 together with the associated cases κ = p - 2 and κ = p - 3.

Original language | English (US) |
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Pages (from-to) | 64-75 |

Number of pages | 12 |

Journal | Fibonacci Quarterly |

Volume | 45 |

Issue number | 1 |

State | Published - Feb 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Fibonacci Quarterly*,

*45*(1), 64-75.