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ON SOLUTIONS OF THE DIOPHANTINE EQUATION $ Q_{\mathfrak{n}}-Q_{\mathfrak{m}}=2^{\mathfrak{a}} $
Keywords:
Pell-Lucas numbers, Diophantine equation, algebraic numbersDOI:
https://doi.org/10.17654/0972555522016Abstract
For the Diophantine equation $Q_{\mathfrak{n}}-Q_{\mathfrak{m}}=2^{\mathfrak{a}}$, where $Q_{\mathfrak{n}}$ and $Q_{\mathfrak{m}}$ are the Pell-Lucas numbers, we find all the non-negative integral solutions in $(\mathfrak{n}, \mathfrak{m}, \mathfrak{a})$. By using the theory of linear forms in logarithms of algebraic numbers, we derive an explicit upper bound for $\mathfrak{n}$, and then, by reduction method based on continued fraction algorithm, we achieve our objective.
Received: January 12, 2022
Accepted: March 27, 2022
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