Open Access

On the appearance of primes in linear recursive sequences

Advances in Difference Equations20052005:868367

DOI: 10.1155/ADE.2005.145

Received: 16 August 2004

Published: 31 May 2005

Abstract

We present an application of difference equations to number theory by considering the set of linear second-order recursive relations, https://static-content.springer.com/image/art%3A10.1155%2FADE.2005.145/MediaObjects/13662_2004_Article_1038_IEq1_HTML.gif , U0 = 0, U1 = 1, and https://static-content.springer.com/image/art%3A10.1155%2FADE.2005.145/MediaObjects/13662_2004_Article_1038_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FADE.2005.145/MediaObjects/13662_2004_Article_1038_IEq3_HTML.gif , 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 2 n p ± 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.

Authors’ Affiliations

(1)
Department of Math & Computer Science, Austin College

Copyright

© Hindawi Publishing Corporation 2005