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On the appearance of primes in linear recursive sequences

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  • Roger Lee Bagula
    After a nice article in this month s American Mathematical Monthly I found this article online that downloads:
    Message 1 of 1 , Oct 23, 2007
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      After a nice article in this month's American Mathematical Monthly
      I found this article online that downloads:

      http://www.hindawi.com/GetArticle.aspx?doi=10.1155/ADE.2005.145
      http://www.hindawi.com/GetPDF.aspx?doi=10.1155/ADE.2005.145
      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



      John H. Jaroma


      Department of Math & Computer Science, Austin College, Sherman 75090,
      TX, USA





      Received 16 August 2004; Revised 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−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.






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