Erazo, Harold;Gómez, Carlos A.;Luca, Florian

Ramanujan J.53, No. 1, 123-137 (2020).

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: thefiniteness result, which is the main result, in which they effectively bound the variables and hence the number of solutions to (3); and thenumerical 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 ofC. 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 inMathematica.

Keywords:

Pell equations;linear combinations of prime powers;lower bounds for linear forms in logarithms