Linear combinations of prime powers in \(X\)-coordinates of Pell equations
type: Item
instance of: scholarly article
title: Linear combinations of prime powers in \(X\)-coordinates of Pell equations
zbMATH Open document ID: 1485.11027
author: Harold Erazo
author: Florian Luca
author: Carlos Alexis Gómez
published in: The Ramanujan Journal
publication date: 2021-05-04T00:00:00Z
review text: 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, th…
reviewed by: Mahadi Ddamulira
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 7343717
zbMATH Keywords: Pell equations
zbMATH Keywords: linear combinations of prime powers
zbMATH Keywords: lower bounds for linear forms in logarithms
MaRDI profile type: Publication
full work available at URL: s11139-019-00213-5
OpenAlex ID: W3009949508
cites work: THE EQUATIONS 3x2−2 = y2 AND 8x2−7 = z2
cites work: Linear combinations of prime powers in binary recurrence sequences
cites work: On the $x$-coordinates of Pell equations which are Fibonacci numbers II
cites work: $X$-coordinates of Pell equations as sums of two tribonacci numbers
cites work: Classical and modular approaches to exponential Diophantine equations. I: Fibonacci and Lucas perfect powers
cites work: Number Theory
cites work: On the \(x\)-coordinates of Pell equations which are \(k\)-generalized Fibonacci numbers
cites work: Q5417755
cites work: On the $x$-coordinates of Pell equations which are rep-digits
cites work: Q4215652
cites work: On $X$-coordinates of Pell equations which are repdigits
cites work: Zeckendorf representations with at most two terms to \(x\)-coordinates of Pell equations
cites work: Distinct digits in basebexpansions of linear recurrence sequences
cites work: Fibonacci numbers of the form \(p^a\pm p^b\)
cites work: Q3598632
cites work: On the $x$-coordinates of Pell equations which are Fibonacci numbers
cites work: On the \(x\)-coordinates of Pell equations which are products of two Fibonacci numbers
cites work: Fibonacci and Lucas numbers of the form \(2^a+3^b+5^c\)
cites work: An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II
cites work: Linear combinations of prime powers in sums of terms of binary recurrence sequences
cites work: Perfect powers in second order linear recurrences
cites work: Q3137208
cites work: Q4280150
cites work: On the Diophantine equation $ax^{2t}+bx^ty+cy^2=d$ and pure powers in recurrence sequences.
cites work: On the representation of an integer in two different bases.
DOI: 10.1007/S11139-019-00213-5
description: scientific article; zbMATH DE number 7343717