Almost powers in the Lucas sequence (Q1026975)

The \textit{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 \(L_3=4\) [\textit{Y. Bugeaud, M. Mignotte} and \textit{S. Siksek}, Ann. Math. (2) 163, No. 3, 969--1018 (2006; Zbl 1113.11021)]. 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 \textit{Modular Method} and the \textit{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. 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 \(L_{19}=9349\). 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''. 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! 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. Reviewer's remark: The authors resort to the routines of \texttt{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 \texttt{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-Pethő-Zimmer) probably because \texttt{MAGMA}'s handbook itself neglects to mention the relevant papers (unlike the case of most its routines).