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

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


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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