Advances in Difference Equations
Volume 2005 (2005), Issue 2, Pages 145-151
doi:10.1155/ADE.2005.145
On the appearance of primes in linear recursive sequences
Department of Math & Computer Science, Austin College, Sherman 75090, TX, USA
Received 16 August 2004; Revised 5 December 2004
Copyright © 2005 John H. Jaroma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We present an application of difference equations to number theory by considering the set of linear second-order recursive relations, Un+2(R,Q)=RUn+1−QUn, U0=0, U1=1, and Vn+2(R,Q)=RVn+1−QVn, V0=2, V1=R, 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.