Year: 2026
Type: article
Abstract: 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 $$k \ge 2$$ 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 $$a_1 a 1 < a 2 < a 3 < a 4 such that all pairwise products $$a_ia_j+1$$ a i a j + 1 (for $$i\ne j$$ i ≠ j ) belong to the k -Pell sequence for any $$k\ge 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\ge 2$$ k ≥ 2 .
Source: Vietnam Journal of Mathematics
Authors: Andrés E. Dorado, Jhon J. Bravo
Institution: University of Cauca
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Funder: Universidad del Cauca
OpenAlex ID: W7160890126