Primefree shifted Lucas sequences of the second kind

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• 作者：Dan Ismailescu^a ; ^{Dan.P.Ismailescu@hofstra.edu} ; Lenny Jones^b ; ^{lkjone@ship.edu} ; Tristan Phillips^b ; ^{tp7924@ship.edu}
• 关键词：primary ; 11B37 ; 11B39 ; secondary ; 11B83
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：173
• 期：Complete
• 页码：87-99
• 全文大小：320 K
• 卷排序：173

文摘

We say a sequence S=(sn)n≥0S=(sn)n≥0 is primefree   if |sn||sn| is not prime for all n≥0n≥0 and, to rule out trivial situations, we require that no single prime divides all terms of SS. Recently, the second author showed that there exist infinitely many integers k   such that both of the shifted sequences U±kU±k are simultaneously primefree, where UU is a particular Lucas sequence of the first kind. In this article, we prove an analogous result for the Lucas sequences Va=(vn)n≥0Va=(vn)n≥0 of the second kind, defined byv0=2,v1=a,andvn=avn−1+vn−2,for n≥2, where a is a fixed integer. More precisely, we show that for any integer a, there exist infinitely many integers k   such that both of the shifted sequences Va±kVa±k are simultaneously primefree. This result provides additional evidence to support a conjecture of Ismailescu and Shim. Moreover, we show that there are infinitely many values of k   such that every term of both of the shifted sequences Va±kVa±k has at least two distinct prime factors.