Diophantine Quadruples with Values in Generalized Pell Sequences

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 $$k \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . 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 $$a_1&lt;a_2&lt;a_3&lt;a_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> </mml:math> such that all pairwise products $$a_ia_j+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> (for $$i\ne j$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>≠</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:math> ) belong to the k -Pell sequence for any $$k\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . 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 $$k\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> .