Equações diofantinas envolvendo sequências de fibonacci generalizadas

The famous and widely studied Fibonacci sequence is determined by the recurrence Fn = Fn−1 + Fn−2, where F0 = 0 and F1 = 1. We can extend this sequence for higher order recurrences. So, for k ≥ 2 and n ≥ −(k − 2), let F (k) n = F (k) n−1 + · · · + F (k) n−k, where F (k) −(k−2) = · · · = F (k) −1 = F (k) 0 = 0 and F1 = 1. We shall study some Diophantine equations involving such sequences. First, we recall that a perfect number is a natural number which equals the sum of all its proper divisors. Then, we shall apply linear forms in logarithms to find even perfect numbers in genereralized Fibonacci sequences. In other words, we shall study the Diophantine equation F (k) n = 2p−1(2p − 1). In another problem, we shall study the 2− adic valuation of F (k) n , when k = 4, in order to find factorials in that sequence, i.e., we shall study the equation Qn = m!. Also, we shall use similar techniques to solve a particular case of the Brocard-Ramanujan equation, n = m! + 1, when the integer n is a number of the previously mentioned sequence.