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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Motivated by the works of Gómez and Luca (Glas. Mat. III 50, 17–24, <CitationRef CitationID="CR12">2015</CitationRef>; Math. Slovaca 68, 939–949, <CitationRef CitationID="CR13">2018</CitationRef>) on Diophantine quadruples with values in generalized Fibonacci sequences, we investigate whether there exist quadruples of positive integers <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a_1<a_2<a_3<a_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo><</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo><</mo> <msub> <mi>a</mi> <mn>3</mn> </msub> <mo><</mo> <msub> <mi>a</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> such that all pairwise products <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$a_ia_j+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>a</mi> <mi>j</mi> </msub> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> (for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$i\ne j$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> </math></EquationSource> </InlineEquation>) belong to the k-Pell sequence for any <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

Diophantine Quadruples with Values in Generalized Pell Sequences

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2026年5月12日

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Andrés E. Dorado,
Jhon J. Bravo

摘要

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 $k \ge 2$ $k \geq 2$ . 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 $a_1<a_2<a_3<a_4$ $a_{1} < a_{2} < a_{3} < a_{4}$ such that all pairwise products $a_ia_j+1$ $a_{i} a_{j} + 1$ (for $i\ne j$ $i \neq j$ ) belong to the k-Pell sequence for any $k\ge 2$ $k \geq 2$ . 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 $k\ge 2$ $k \geq 2$ .