Abstract
For the Fibonacci sequence the identity \(F_n^2+F_{n+1}^2 = F_{2n+1}\) holds for all \(n \ge 0\). Let \({\mathcal {X}}:= (X_\ell )_{\ell \ge 1}\) be the sequence of X-coordinates of the positive integer solutions (X, Y) of the Pell equation \(X^2-dY^2=\pm 1\) corresponding to a nonsquare integer \(d>1\). In this paper, we investigate all positive nonsquare integers d for which there are at least two positive integers X and \(X'\) of \(\mathcal {X}\) having a representation as the sum of xth powers of two consecutive terms of a Lucas sequence. Then we solve this problem for Fibonacci numbers.
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Notes
The k-generalized Fibonacci sequence \(F^{(k)}\), for an integer \(k\ge 2\), satisfies that its first k terms are \(0, \ldots , 0, 1\) and each term afterwards is the sum of the preceding k terms. For \(k = 2\), this reduces to the familiar Fibonacci numbers, while for \(k = 3\) these are the Tribonacci numbers. They are followed by the Tetranacci numbers for \(k = 4\), and so on.
For a positive integer base \(b \ge 2\), a repdigit is a positive integer N whose base b-representation has a unique repeating digit.
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Acknowledgements
We thank the referee for a careful reading of the manuscript and for several suggestions which improved the presentation of our paper. The first author was partially supported by CAPES while C. A. G. was supported in part by Project 71280 (Universidad del Valle).
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Erazo, H.S., Gómez, C.A. An exponential Diophantine equation related to the sum of powers of two consecutive terms of a Lucas sequence and x-coordinates of Pell equations. Period Math Hung 83, 165–184 (2021). https://doi.org/10.1007/s10998-021-00388-9
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DOI: https://doi.org/10.1007/s10998-021-00388-9