Sobre somas de potências de termos consecutivos na sequência de Fibonacci k-generalizada

• Main Author: Rico Acevedo, Carlos Alirio
• Publication Date: 2018
• Format: Master thesis
• Language: por
• Source: Repositório Institucional da UFG
• Download full: http://repositorio.bc.ufg.br/tede/handle/tede/8329
• topic: Equações diofantinas; Sequência de fibonacci k-generalizada; Formas lineares em logaritmos; Metodos de redução; Diphantine equation; K-generalized fibonacci sequence; Linear form in logarithm; Reduction method; ALGEBRA::TEORIA DOS NUMEROS

Let $ k \geq 2.$ an integer. The recurrence $ \fk{n} = \sum_ {i = 0}^k \fk{n-i} $ for $ n> k $, with initial conditions $F_{-(k-2)}^{(k)}=F_{-(k-3)}^{(k)}=\cdots=F_{0}^{(k)}=0$ and $F_1^{ (k)} = 1$, which is called the $k$-generalized Fibonacci sequence. When $ k = 2 ,$ we have the Fibonacci sequence $ \{ F_n \}_{n\geq 0}.$ We will show that the equation $F_{n}^{x}+F_{n+1}^x=F_{m}$ does not have no non-trivial integer solutions $ (n, m, x) $ to $ x> 2 $. On the other hand, for $ k \geq 3,$ we will show that the diophantine equation $\epi$ does not have integer solutions $ (n, m, k, x) $ with $ x \geq 2 $. In both cases, we will use initially Matveev's Theorem, for linear forms in logarithms and the reduction method due to Dujella and Pethö, to limit the variables $ n, \; m $ and $ x $ at intervals where the problem is computable. In addition, in the case for $ k\geq 3 $, we will use the fact that the dominant root the $k$-generalized Fibonacci sequence is exponentially close to 2 to bound $k$, a method developed by Bravo and Luca.