On solutions of the Diophantine equation $L_n+L_m=3^a$

Abstract

Let $(L_n{)}_{n\ge 0}$ be the Lucas sequence given by $L_0=2,L_1=1$ and $L_{n+2}=L_{n+1}+L_n$ for $n\ge 0$. In this paper, we are interested in finding all powers of three which are sums of two Lucas numbers, i.e., we study the exponential Diophantine equation $L_n+L_m=3^a$ 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.

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