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

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 (*L*_{n})_{n ≥ 0}
be the Lucas sequence given by *L*_{0} = 0,
*L*_{1} = 1,
and *L*_{n+2} =
*L*_{n+1} +
*L*_{n} 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
*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.

(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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