Geodesic
Powers of two as sums of two Lucas numbers
Journal of integer sequences, Tome 17 (2014) no. 8.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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},
     publisher = {mathdoc},
     volume = {17},
     number = {8},
     year = {2014},
     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/