Journal of Integer Sequences, Vol. 17 (2014), Article 14.8.3

Powers of Two as Sums of Two Lucas Numbers


Jhon J. Bravo
Mathematics Department
University of Cauca
Street 5 No. 4-70
Popayán, Cauca
Colombia

Florian Luca
School of Mathematics
University of the Witwatersrand
P. O. Box Wits 2050
South Africa
and
Mathematical Institute
UNAM Juriquilla
Santiago de Querétaro 76230
Querétaro de Arteaga
Mexico

Abstract:

Let (Ln)n ≥ 0 be the Lucas sequence given by L0 = 0, L1 = 1, and Ln+2 = Ln+1 + Ln for n ≥ 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 Ln + Lm = 2a 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.


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(Concerned with sequences A000032 A000045.)


Received March 17 2014; revised versions received July 22 2014; July 30 2014. Published in Journal of Integer Sequences, July 30 2014.


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