Abstract
Let \((u_{n})_{n \ge 0}\) be a non-degenerate binary recurrence sequence with positive discriminant and p be a fixed prime number. In this paper, we are interested in finding a finiteness result for the solutions of the Diophantine equation \(u_{n_{1}} + u_{n_{2}} + \cdots + u_{n_{t}} = p^{z}\) with \(n_1> n_2> \cdots > n_t\ge 0\). Moreover, we explicitly find all the powers of three which are sums of three balancing numbers using lower bounds for linear forms in logarithms. Further, we use a variant of the Baker–Davenport reduction method in Diophantine approximation due to Dujella and Pethő.
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Acknowledgements
We thank the referee for suggestions which improved the quality of this paper. The first author would like to thank Harish-Chandra Research Institute, Allahabad and Institute of Mathematics & Applications, Bhubaneswar for their warm hospitality during the academic visits.
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Communicated by J. Schoißengeier.
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Mazumdar, E., Rout, S.S. Prime powers in sums of terms of binary recurrence sequences. Monatsh Math 189, 695–714 (2019). https://doi.org/10.1007/s00605-019-01282-w
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DOI: https://doi.org/10.1007/s00605-019-01282-w