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On solutions of the Diophantine equation $L_n+L_m=3^a$ (2202.13182v1)
Published 26 Feb 2022 in math.NT
Abstract: Let $(L_n){n\geq 0}$ be the Lucas sequence given by $L_0 = 2, L_1 = 1$ and $L{n+2} = L_{n+1}+L_n$ for $n \geq 0$. 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.