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.
Highlights
We move our interest on the powers of 3 as a sum of two Lucas numbers
We are interested in finding all powers of three which are sums of two Lucas numbers, i.e., we study the exponential Diophantine equation Ln + Lm = 3a in nonnegative integers n, m, and a
This paper follows the following steps : We first give the generalities on binary linear recurrence, we demonstrate an important inequality on Lucas numbers and determine and reduce a coarse bound by section 3
Summary
We suppose that ak = 0 (otherwise, the sequence {Hn}n≥0 satisfies a recurrence of order less than k). If k = 2, the sequence (Hn)n≥0 is called a binary recurrent sequence. In this case, the characteristic polynomial is of the form f (X) = X2 − a1X − a2 = (X − α1)(X − α2). The binary recurrent sequence {Hn}n≥0 is said to be non degenerated if c1c2α1α2 = 0 and α1/α2 is not a root of unity. Let us define another height, deduced from the last one, called the absolute logarithmic height
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