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Open Access Open Badges Research Article

On the appearance of primes in linear recursive sequences

John H Jaroma

Author Affiliations

Department of Math & Computer Science, Austin College, Sherman, TX 75090, USA

Advances in Difference Equations 2005, 2005:868367  doi:10.1155/ADE.2005.145

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2005/2/868367

Received:16 August 2004
Revisions received:5 December 2004
Published:31 May 2005

© 2005 Jaroma

We present an application of difference equations to number theory by considering the set of linear second-order recursive relations, , U0 = 0, U1 = 1, and , , where R and Q are relatively prime integers and n ∈ {0,1,...}. These equations describe the set of extended Lucas sequences, or rather, the Lehmer sequences. We add that the rank of apparition of an odd prime p in a specific Lehmer sequence is the index of the first term that contains p as a divisor. In this paper, we obtain results that pertain to the rank of apparition of primes of the form 2np ± 1. Upon doing so, we will also establish rank of apparition results under more explicit hypotheses for some notable special cases of the Lehmer sequences. Presently, there does not exist a closed formula that will produce the rank of apparition of an arbitrary prime in any of the aforementioned sequences.

Research Article