Complete padovan sequences in finite fields

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)64-75
Number of pages12
JournalFibonacci Quarterly
Volume45
Issue number1
StatePublished - Feb 2007

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Galois field
Primitive Roots
Linear Recurrence

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

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title = "Complete padovan sequences in finite fields",
abstract = "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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Complete padovan sequences in finite fields. / Gil, Juan Bautista; Weiner, Michael David; Zara, Catalin.

In: Fibonacci Quarterly, Vol. 45, No. 1, 02.2007, p. 64-75.

Research output: Contribution to journalArticle

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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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