A Quadratic Diophantine Equation Involving Generalized Fibonacci Numbers

Abstract

The sequence of the k-generalized Fibonacci numbers (F-n((k)))(n) is defined by the recurrence F-n((k)) = Sigma(k)(j) = 1 F-n-j((k)) beginning with the k terms 0,..., 0, 1. In this paper, we shall solve the Diophantine equation F-n((k)) = (F-m((l)))(2) + 1, in positive integers m, n, k and l.