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Item Equações diofantinas envolvendo a soma de quadrados de números de Fibonacci k-generalizada(Universidade Federal de Goiás, 2019-12-20) Carvalho, Camila Santos de Sá; Chaves, Ana Paula de Araújo; http://lattes.cnpq.br/2332073500640724; Chaves, Ana Paula de Araújo; Lima, Lidiane dos Santos Monteiro; Ferreira, Diego MarquesFibonaccinumbers(Fn)n, where F0 = 0, F1 =1 and Fn+2 =Fn+1+Fn forn ≥ 0, hasseveral generalizations. Among them, we have the sequence (F(k) n )n, given by F(k) n = F(k) n−1 + ···+F(k) n+k, for every n≥2, with initial values F(k) −(k−2) = F(k) −(k−3) =···= F(k) 0 = 0 and F(k) 1 = 1, which is called the k-generalized Fibonacci sequence (or k-bonacci sequence). Inspired by the identity F2 n +F2 n+1 = F2n+1, which tells us that the sum of squares of two consecutive Fibonacci numbers is also a Fibonacci number, Chaves and Marques number, in 2014, showed that the Diophantine equation (F(k) n )2+(F(k) n+1)2 = F(k) m has no solutions in positive integers n,m and k, with n > 1 and k≥3, which means that the mentioned identity is not satisfied for k-bonacci numbers, outside the initial values. In this work, based on the paper of Bednaˇrík, Freitas, Marques and Trojovský (2019), we will show that the Diophantine equation (F(k) n )2 +(F(k) n+1)2 = F(l) m , has no solution to 2≤k < l e n > k+1, implying that the sum of squares of two consecutive k - bonacci numbers does not belong to another l-generalized Fibonacci sequence of greater order.Item Sobre somas de potências de termos consecutivos na sequência de Fibonacci k-generalizada(Universidade Federal de Goiás, 2018-03-16) Rico Acevedo, Carlos Alirio; Chaves, Ana Paula de Araújo; http://lattes.cnpq.br/2332073500640724; Chaves, Ana Paula de Araújo; Martinez, Fabio Enrique Brochero; Oliveira, Ricardo NunesLet $ 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.Item Altura e equidistribuição de pontos algébricos(Universidade Federal de Goiás, 2017-06-20) Santos, Jefferson Marques; Chaves, Ana Paula de Araújo; http://lattes.cnpq.br/2332073500640724; Chaves , Ana Paula de Araújo; Lopes , José Othon Dantas; Rodrigues , Paulo Henrique de Azevedo; Oliveira, Ricardo Nunes deThe concept of roots of a polynomial is quite simple but has several applications. This concept extends more generally to the case of "small" algebraic points sequences in a curve. This dissertation aims to estimate the size of algebraic numbers by means of Weil height. In addition to showing that they are distributed evenly around the unit circle, through Bilu Equidistribution Theorem.