On the $x$--coordinates of Pell equations which are $k$--generalized Fibonacci numbers
Mahadi Ddamulira,Florian Luca +1 more
TLDR
For an integer d ≥ 2 which is square-free, this paper showed that there is at most one value of the positive integer x participating in the Pell equation x 2 − d y 2 = ± 1, which is a k-generalized Fibonacci number, with a couple of parametric exceptions which they completely characterize.About:
This article is published in Journal of Number Theory.The article was published on 2020-02-01 and is currently open access. It has received 11 citations till now. The article focuses on the topics: Fibonacci number & Pell's equation.read more
Citations
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Linear combinations of prime powers in X -coordinates of Pell equations
TL;DR: In this paper, it was shown that there are only a finite number of nonsquare integers that can be represented as a linear combination of prime powers with fixed primes and coefficients, restricted to the condition that the exponent of the largest prime is the greatest of all exponents.
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On the $x-$coordinates of Pell equations which are sums of two Padovan numbers
TL;DR: In this article, it was shown that all positive square-free integers have at least two positive integer solutions, such that each of these solutions is a sum of two Padovan numbers.
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The x-coordinates of Pell equations and sums of two Fibonacci numbers II
TL;DR: In this paper, it was shown that there is at most one value of the positive integer x participating in the Pell equation, which is a sum of two Fibonacci numbers, with a few exceptions that are completely characterized.
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On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences
TL;DR: The generalized k-distance Tribonacci sequence (Tn(k))n≥0 as mentioned in this paper is a generalization of the Wloch sequence, which was introduced in 2008.
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Sums of S-units in the solution sets of generalized Pell equations
Lajos Hajdu,P. Sebestyén +1 more
TL;DR: In this article, the authors give various finiteness results concerning solutions of generalized Pell equations representable as sums of S-units with a fixed number of terms, and in case of more terms, they are able to bound the number of solutions.
References
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An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers
TL;DR: In this paper, the authors studied linear forms with rational integer coefficients satisfying the strong independence condition and proved an explicit estimate of the form in standard notation, and showed that the form contains no factors in the form.
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A Generalization of a Theorem of Baker and Davenport
Andrej Dujella,Attila Pethö +1 more
TL;DR: In this article, the authors generalize the result of Baker and Davenport and prove that the Diophantine pair {1, 3, 8, d} cannot be extended to infinitely many Diophantus quadruples.
Journal ArticleDOI
Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers
TL;DR: In this paper, the authors combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's last theorem.
Related Papers (5)
On the $x$--coordinates of Pell equations which are $k$--generalized Fibonacci numbers
Mahadi Ddamulira,Florian Luca +1 more
On the $x$-coordinates of Pell equations which are Fibonacci numbers
Florian Luca,Alain Togbé +1 more