Linear combinations of prime powers in \(X\)-coordinates of Pell equations
Let \( d>1 \) be a positive squarefree integer. It is well-known that the Pell equation \[X^2-dY^2=\pm 1,\tag{1}\] has infinitely may positive integer solutions \( (X,Y) \), of the form \[X+Y\sqrt{d}=X_k+Y_k\sqrt{d}=(X_1+Y_1\sqrt{d})^{k},\tag{2}\] for some positive integer \( k \), where \( (X_1,Y_1) \) is the smallest positive integer solution of (1). Furthermore, by conjugating the relation in (2), one obtains the Binet formula for the binary recurrent sequence \( \{X_k\}_{k\ge 1} \) given by \begin{align*} X_k=\dfrac{(X_1+Y_1\sqrt{d})^{k}+(X_1-Y_1\sqrt{d})^{k}}{2}. \end{align*} In the paper under review, the authors investigate the following problem. For \( s \) a positive integer, they determine for which positive squarefree integers \( d>1 \) the sequence \( \{X_\ell\}_{\ell\ge 1} \) has at least two different terms that can be represented as \[X_{\ell}=c_1p_1^{n_1}+\cdots+c_sp_s^{n_s}, \tag{3}\] where \( \epsilon \in (0,1) \), \( c_1, \ldots, c_s\in \mathbb{Z}^{+};~ p_1, \ldots, p_s \) are primes with \( p_1\le \cdots \le p_s \); \( n_1, \ldots, n_s\ge 0 \); and \( (1-\epsilon)n_s => \max_{1\le i\le s}n_i> \max_{1\le i\le s-1} n_i \). In other words, if \( H_{d,\epsilon} \) is the set of solutions \( (\ell, n_1, \ldots, n_s) \) of (3), the authors are interested in determining the values of \( d \) for which there exist two solutions \( (\ell_1, a_1, \ldots, a_s) \), \( (\ell_2, b_1, \ldots, b_s) \) both in \( H_{d,\epsilon} \), with \( \ell_1\ne \ell_2 \). The authors prove two interesting results: the \textbf{finiteness result}, which is the main result, in which they effectively bound the variables and hence the number of solutions to (3); and the \textbf{numerical result}, in which they effectively solve a particular case of (3), that is, \( X_\ell=2^{n_1}+3^{n_2}\), with \( n_1 \le n_2 \). Thier research is motivated by the result of \textit{C. Bertók} et al. in [Int. J. Number Theory 13, No. 2, 261--271 (2017; Zbl 1409.11009)]. To prove their results, the authors use a clever combination of techniques in number theory, the properties of the solutions of Pell equations, the theory of nonzero linear forms in logarithms of algebraic numbers `á la Baker', and the reduction techniques involving the theory of continued fractions. All computations are done with the help of a computer program in \texttt{Mathematica}.
lower bounds for linear forms in logarithms
scientific article; zbMATH DE number 7343717
scientific article