Report
On the $x$--coordinates of Pell equations which are $k$--generalized Fibonacci numbers
| العنوان: | On the $x$--coordinates of Pell equations which are $k$--generalized Fibonacci numbers |
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| المؤلفون: | Ddamulira, Mahadi, Luca, Florian |
| سنة النشر: | 2018 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Number Theory, 11A25, 11B39, 11J86 |
| الوصف: | For an integer $k\geq 2$, let $\{F^{(k)}_{n}\}_{n\geqslant 2-k}$ be the $ k$--generalized Fibonacci sequence which starts with $0, \ldots, 0,1$ (a total of $k$ terms) and for which each term afterwards is the sum of the $k$ preceding terms. In this paper, for an integer $d\geq 2$ which is square free, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^{2}-dy^{2} =\pm 1$ which is a $k$--generalized Fibonacci number, with a couple of parametric exceptions which we completely characterise. This paper extends previous work from [17] for the case $k=2$ and [16] for the case $k=3$. Comment: 36 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/1803.10434 |
| رقم الانضمام: | edsarx.1803.10434 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |