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<p>The Padovan (<i>P</i><sub><i>n</i></sub>)<sub><i>n</i>≥0</sub> and Perrin (<i>R</i><sub><i>n</i></sub>)<sub><i>n</i>≥0</sub> sequences are third-order linear recurrences, both defined by the relation <i>u</i><sub><i>n</i></sub> = <i>u</i><sub><i>n−</i>2</sub>+<i>u</i><sub><i>n−</i>3</sub> for <i>n</i> ≥ 3. They differ in their initial conditions resulting in different sequences. The Padovan sequence begins with <i>P</i><sub>0</sub> = <i>P</i><sub>1</sub> = <i>P</i><sub>2</sub> = 1, whereas the Perrin sequence starts with <i>R</i><sub>0</sub> = 3, <i>R</i><sub>1</sub> = 0<i>,</i> and <i>R</i><sub>2</sub> = 2. Motivated by the work of Gómez and Luca [Tribonacci Diophantine quadruples, <i>Glas. Mat., Ser. III</i>, 50(1):17–24, 2015], we investigate whether there exist quadruples of positive integers <i>a</i><sub>1</sub> <i>&lt; a</i><sub>2</sub> <i>&lt; a</i><sub>3</sub> <i>&lt; a</i><sub>4</sub> such that all pairwise products <i>a</i><sub><i>i</i></sub><i>a</i><sub><i>j</i></sub> + 1 (for <i>i ≠ j</i>) belong to the Padovan or Perrin sequence, and we prove that the answer is negative.</p>

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Diophantine quadruples with values in the Padovan and Perrin sequences

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  • 发表日期：

    2026年1月1日

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  • Andrés E. Dorado,
  • Jhon J. Bravo

摘要

The Padovan (Pn)n≥0 and Perrin (Rn)n≥0 sequences are third-order linear recurrences, both defined by the relation un = un−2+un−3 for n ≥ 3. They differ in their initial conditions resulting in different sequences. The Padovan sequence begins with P0 = P1 = P2 = 1, whereas the Perrin sequence starts with R0 = 3, R1 = 0, and R2 = 2. Motivated by the work of Gómez and Luca [Tribonacci Diophantine quadruples, Glas. Mat., Ser. III, 50(1):17–24, 2015], we investigate whether there exist quadruples of positive integers a1 < a2 < a3 < a4 such that all pairwise products aiaj + 1 (for i ≠ j) belong to the Padovan or Perrin sequence, and we prove that the answer is negative.