On the Appearance of Primes in Linear Recursive Sequences

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.