کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
8897167 | 1630641 | 2017 | 13 صفحه PDF | دانلود رایگان |
عنوان انگلیسی مقاله ISI
Primefree shifted Lucas sequences of the second kind
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موضوعات مرتبط
مهندسی و علوم پایه
ریاضیات
اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
چکیده انگلیسی
We say a sequence S=(sn)nâ¥0 is primefree if |sn| is not prime for all nâ¥0 and, to rule out trivial situations, we require that no single prime divides all terms of S. Recently, the second author showed that there exist infinitely many integers k such that both of the shifted sequences U±k are simultaneously primefree, where U is a particular Lucas sequence of the first kind. In this article, we prove an analogous result for the Lucas sequences Va=(vn)nâ¥0 of the second kind, defined byv0=2,v1=a,andvn=avnâ1+vnâ2,for nâ¥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±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±k has at least two distinct prime factors.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Number Theory - Volume 173, April 2017, Pages 87-99
Journal: Journal of Number Theory - Volume 173, April 2017, Pages 87-99
نویسندگان
Dan Ismailescu, Lenny Jones, Tristan Phillips,