On the appearance of primes in linear recursive sequences

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.