Báo Cáo Báo cáo hóa học ON THE APPEARANCE OF PRIMES IN LINEAR RECURSIVE SEQUENCES JOHN H JAROMA

ON THE APPEARANCE OF PRIMES IN LINEAR RECURSIVE SEQUENCES JOHN H. JAROMA Received 16 August 2004 and in revised form 5 December 2004

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 ư QUn , U0 = 0, U1 = 1, √ √ and Vn+2 ( R,Q) = RVn+1 ư 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 2n 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