在Pell方程的x坐标上它是两个Lucas数的乘积

IF 0.4 Q4 MATHEMATICS FIBONACCI QUARTERLY Pub Date : 2020-02-01 DOI:10.33774/COE-2020-27J3Q

Mahadi Ddamulira

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引用次数: 5

摘要

设$ \{L_n\}_{n\ge 0} $为所有$ n\ge 0 $由$ L_0=2, ~ L_1=1 $和$ L_{n+2}=L_{n+1}+L_n $给出的卢卡斯数序列。在本文中，对于一个无平方的整数$d\geq 2$，我们证明了Pell方程$x^{2}-dy^{2}=\pm 1$中最多有一个正整数$x$的值，它是两个Lucas数的乘积，除了一些我们完全描述的例外。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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On the $x$--coordinates of Pell equations that are products of two Lucas numbers

Let $ \{L_n\}_{n\ge 0} $ be the sequence of Lucas numbers given by $ L_0=2, ~ L_1=1 $ and $ L_{n+2}=L_{n+1}+L_n $ for all $ n\ge 0 $. 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 product of two Lucas numbers, with a few exceptions that we completely characterize.

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