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

TL;DR

This paper investigates all solutions to the equation where the sum of two Lucas numbers equals a power of three, employing advanced number theory techniques to find all such solutions.

Contribution

It provides a complete characterization of solutions to the exponential Diophantine equation involving Lucas numbers and powers of three, using bounds and reduction methods.

Findings

Identifies all solutions to $L_n + L_m = 3^a$ in nonnegative integers.

Employs linear forms in logarithms and continued fractions for bounds.

Uses Baker-Davenport reduction to refine solutions.

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.