Diophantine Quadruples with Values in Generalized Pell Sequences

The k -Pell sequence is a generalization of the classical Pell sequence obtained by extending the order of its defining linear recurrence from the second order to an arbitrary order k ≥ 2 k ≥ 2 . Motivated by the works of Gómez and Luca (Glas. Mat. III 50, 17–24, 2015; Math. Slovaca 68, 939–949, 2018) on Diophantine quadruples with values in generalized Fibonacci sequences, we investigate whether there exist quadruples of positive integers a₁<a₂<a₃<a₄ a 1 < a 2 < a 3 < a 4 such that all pairwise products aᵢaⱼ+1 a i a j + 1 (for i≠ j i ≠ j ) belong to the k -Pell sequence for any k≥ 2 k ≥ 2 . In this paper, we prove that no such Diophantine quadruples exist, thus showing that there are no Diophantine quadruples with values in the k -Pell sequence for any k≥ 2 k ≥ 2 .