Equações diofantinas envolvendo a soma de quadrados de números de Fibonacci k-generalizada - Carvalho, Camila Santos de Sá (2019)
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.