Diophantine quadruples with values in k-generalized Fibonacci numbers

Abstract

Abstract We consider for integers k ≥ 2 the k–generalized Fibonacci sequences F (k) := ( F n ( k ) ) n ≥ 2 − k , $\begin{array}{} (F_n^{(k)})_{n\geq 2-k}, \end{array} $ whose first k terms are 0, …, 0, 1 and each term afterwards is the sum of the preceding k terms. In this paper, we show that there does not exist a quadruple of positive integers a 1 < a 2 < a 3 < a 4 such that ai aj + 1 (i ≠ j) are all members of F (k).