Almost powers in the Lucas sequence
Almost powers in the Lucas sequence
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DOI：
10.5802/jtnb.642
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发表时间：
2008
影响因子：
0.4
通讯作者：
Y. Bugeaud;F. Luca;M. Mignotte;S. Siksek
中科院分区：
文献类型：
--
作者：
Y. Bugeaud;F. Luca;M. Mignotte;S. Siksek
卢卡斯序列}$(L_N)_{n\geq0}$定义为：$L_0=2，L_1=1$，$L_n=L_{n-1}+L_{n-2}$对$n\geq 2$。第一、第三和第四作者已经证明了Lucas数列中唯一的完美幂是$L_1=1$和$L_3=4$[{\it Y.Bugeaud，M.米格诺特}和{\it S.Siksek}，Ann]。数学课。(2)第163号，第969-1018号(2006年；zbl 1113.11021)]。本文讨论方程$L_n=q^ay^p$，其中$pq2$和$q$为素数，$a，y$为正整数.近年来，许多有趣的论文致力于类似问题的研究，其中Lucas和/或Fibonacci序列主要表示。这些问题的解决是各种工具的结合，比如初等技巧，所谓的{\it模法}和{\it Double Frey方法}--这两种方法都是在费马大定理的证明之后得到启发和发展的--两个或三个对数的锐界(‘二’或‘三’，这与实际观点有很大的不同！)提到最重要的一点。本文利用这些工具，结合各种减少计算量的技巧，证明了如下定理：$L_n=q^ay^p$的唯一解是$q<；与1087、2207、4481、14503、19207、21503、34303、48767、119809、232049、524287、573569、812167不同的10^6$(在上述关于$p、q、a、y$的假设下)和$q$是：$L_0=2$、$L_2=3$、$L_3=2^2$、$L_4=7$、$L_5=11$、$L_6=2\cot3^2$、$L_7=29$、$L_8=47$、$L_9=19\cot2^2$、$L_{11}=199$、L${13}=521$、L${17}=3 571$和$L_{19}=9349$。根据作者的说法，排除上面给出的13个$Q$值的原因是“对于目前可用的硬件来说，这些值所需的必要的模形式计算要求太高了”。值得注意的是，作者利用初等论据，在计算机的帮助下，证明了(他们的证明绝不是简单的)他们的定理的有效性，除了$Q 3，7，47,127美元。换句话说，所有的重型机械都是为了这四个$Q$的值而使用的，特别是前三个！在这位评论家看来，在这篇论文的其他读者中，对丢番图方程的显式解感兴趣的研究生可以受益良多。这篇论文对他们来说是一本极好的读物，因为它为学习现代数论的美丽“章节”提供了一个有吸引力的动机。评审者的评论：作者求助于{\tt pari/gp}例程来解一些Thue方程，并采取正确的态度，他们不遗漏(在第566页)这些例程所基于的论文(Bilu-Hanrot和Hanrot)。在同一页上，他们还转向{\TT Magma}，以显式计算$Y^2=X(X^2-100q^2)$的所有整数解，其中$q=3，7，47$。不幸的是，他们在这里遗漏了同样的内容(相关例程是基于Stroeker-Tzanakis的论文；Gebel-peth\H{o}-Zimmer)，可能是因为S手册本身忽略了相关论文(与大多数例程不同)。
The {\it Lucas sequence} $(L_n)_{n\geq 0}$ is defined by $L_0=2, L_1=1$ and $L_n=L_{n-1}+L_{n-2}$ for $n\geq 2$. The first, third and fourth authors have proved, among other things, that the only perfect powers in the Lucas sequence are $L_1=1$ and $L_3=4$ [{\it Y. Bugeaud, M. Mignotte} and {\it S. Siksek}, Ann. Math. (2) 163, No. 3, 969--1018 (2006; Zbl 1113.11021)]. \par The present paper deals with the equation $L_n=q^ay^p$, where $p\geq 2$ and $q$ are primes, and $a,y$ are positive integers. In recent years, many interesting papers have been devoted to the study of analogous problems, in which the sequences of Lucas and/or Fibonacci mainly figure. The solution of such problems results from the combination of a variety of tools, like elementary tricks, the so called {\it Modular Method} and the {\it Double Frey Method} -- both inspired by and developed after the proof of Fermat's Last Theorem --, sharp bounds of two or three logarithms (``two'' or ``three'', that makes a big difference from the practical point of view!) to mention the most important. \par The present paper uses all these tools, in combination with various tricks which reduce the amount of computations, in order to prove the following Theorem: The only solutions to $L_n=q^ay^p$ with $q<10^6$ (under the assumptions on $p,q,a,y$ mentioned above) and $q$ different from 1087, 2207, 4481, 14503, 19207, 21503, 34303, 48767, 119809, 232049, 524287, 573569, 812167, are $L_0=2$, $L_2=3$, $L_3=2^2$, $L_4=7$, $L_5=11$, $L_6=2\cdot 3^2$, $L_7=29$, $L_8=47$, $L_9=19\cdot 2^2$, $L_{11}=199$, $L_{13}=521$, $L_{17}=3571$ and $L_{19}=9349$. \par The reason for the exclusion of the thirteen values of $q$ given above is, according to the authors, because ``the necessary modular forms computations needed for these values are too demanding for the currently available hardware''. \par It is worth noting that, using elementary arguments and aided by the computer, the authors prove (their proof being by no means straightforward) the validity of their theorem except for $q\neq 3,7,47,127$. In other words, all the heavy machinery is used ``for the sake'' of these four values of $q$, especially the first three! \par In this reviewer's opinion, among other readers of the paper, graduate students interested in the explicit resolution of Diophantine equations can profit very much. This paper is an excellent reading for them, as it offers an attractive motivation for studying beautiful ``chapters'' of modern Number Theory. \par Reviewer's remark: The authors resort to the routines of {\tt PARI/GP} in order to solve a number of Thue equations and, adopting the right attitude, they do not omit to mention (on page 566) the papers (of Bilu-Hanrot and Hanrot) on which these routines are based. On that same page, they also turn to {\tt MAGMA} in order to explicitly compute all integer solutions to $Y^2=X(X^2-100q^2)$ with $q=3,7,47$. Unfortunately, here they omit to do the same (the relevant routine is based on papers by Stroeker-Tzanakis; Gebel-Peth\H{o}-Zimmer) probably because {\tt MAGMA}'s handbook itself neglects to mention the relevant papers (unlike the case of most its routines).