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arXiv · 2304.11607

On Mixed Concatenations of Pell and Pell-Lucas Numbers which are Pell Numbers

Kouèssi Norbert Adédji·Marija Bliznac Trebješanin
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Abstract

Let $(P_n)_{n\ge 0}$ and $(Q_n )_{n\ge 0}$ be the Pell and Pell-Lucas sequences. Let $b$ be a positive integer such that $b\ge 2.$ In this paper, we prove that the following two Diophantine equations $P_{n}=b^{d}P_{m}+Q_{k}$ and $P_{n}=b^{d}Q_{m}+P_{k}$ with $d,$ the number of digits of $P_k$ or $Q_k$ in base $b,$ have only finitely many solutions in nonegative integers $(m, n, k, b, d)$. Also, we explicitly determine these solutions in cases $2 \le b \le 10.$

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Kouèssi Norbert Adédji, Marija Bliznac Trebješanin. 2023-04-23. On Mixed Concatenations of Pell and Pell-Lucas Numbers which are Pell Numbers. https://doi.org/10.1556/314.2024.00010

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