### 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) |
---|---|

Pages (from-to) | 64-75 |

Number of pages | 12 |

Journal | Fibonacci Quarterly |

Volume | 45 |

Issue number | 1 |

State | Published - Feb 2007 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Fibonacci Quarterly*,

*45*(1), 64-75.

}

*Fibonacci Quarterly*, vol. 45, no. 1, pp. 64-75.

**Complete padovan sequences in finite fields.** / Gil, Juan Bautista; Weiner, Michael David; Zara, Catalin.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Complete padovan sequences in finite fields

AU - Gil, Juan Bautista

AU - Weiner, Michael David

AU - Zara, Catalin

PY - 2007/2

Y1 - 2007/2

N2 - Given a prime p ≥, and given 1 < κ < p-1, we call a sequence (an)n in Fp a Φκ-sequence if it is periodic with period p-1, and if it satisfies the linear recurrence a n + an+1 - an+κ with a0 = 1. Such a sequence is said to be a complete Φκ-sequence if in addition {a0,a1, ⋯, ap-2} = {1, ⋯,p-1}. For instance, every primitive root b mod p generates a complete Φκ-sequence an = bn for some (unique) κ. A natural question is whether every complete Φκ- 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.

AB - Given a prime p ≥, and given 1 < κ < p-1, we call a sequence (an)n in Fp a Φκ-sequence if it is periodic with period p-1, and if it satisfies the linear recurrence a n + an+1 - an+κ with a0 = 1. Such a sequence is said to be a complete Φκ-sequence if in addition {a0,a1, ⋯, ap-2} = {1, ⋯,p-1}. For instance, every primitive root b mod p generates a complete Φκ-sequence an = bn for some (unique) κ. A natural question is whether every complete Φκ- 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.

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M3 - Article

VL - 45

SP - 64

EP - 75

JO - Fibonacci Quarterly

JF - Fibonacci Quarterly

SN - 0015-0517

IS - 1

ER -