Equações diofantinas envolvendo a soma de quadrados de números de Fibonacci k-generalizada
Nenhuma Miniatura disponível
Data
2019-12-20
Autores
Título da Revista
ISSN da Revista
Título de Volume
Editor
Universidade Federal de Goiás
Resumo
Fibonaccinumbers(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.
Descrição
Citação
CARVALHO, Camila Santos de Sá. Equações diofantinas envolvendo a soma de quadrados de números de Fibonacci k-generalizada. 2019. 54 f. Dissertação (Mestrado em Matemática) - Universidade Federal de Goiás, Goiânia, 2019.