Geodesic
Powers of two as sums of two Lucas numbers
Journal of integer sequences, Tome 17 (2014) no. 8
Let $(L_{n})_{n \ge 0}$ be the Lucas sequence given by $L_{0} = 0, L_{1} = 1$, and $L_{n+2} = L_{n+1} + L_{n}$ for $n \ge 0$. In this paper, we are interested in finding all powers of two which are sums of two Lucas numbers, i.e., we study the Diophantine equation $L_{n} + L_{m} = 2^{a}$ in nonnegative integers $n, m$, and $a$. The proof of our main theorem uses lower bounds for linear forms in logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in diophantine approximation. This paper continues our previous work where we obtained a similar result for the Fibonacci numbers.
Classification : 11B39, 11J86
Keywords: Fibonacci number, Lucas number, linear forms in logarithms, continued fraction, reduction method 
@article{JIS_2014__17_8_a0,
     author = {Bravo,  Jhon J. and Luca,  Florian},
     title = {Powers of two as sums of two {Lucas} numbers},
     journal = {Journal of integer sequences},
     year = {2014},
     volume = {17},
     number = {8},
     zbl = {1358.11026},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2014__17_8_a0/}
}
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Bravo,  Jhon J.; Luca,  Florian. Powers of two as sums of two Lucas numbers. Journal of integer sequences, Tome 17 (2014) no. 8. http://geodesic.mathdoc.fr/item/JIS_2014__17_8_a0/