Diophantine quadruples with values in the Padovan and Perrin sequences

The Padovan ( P n ) n ≥0 and Perrin ( R n ) n ≥0 sequences are third-order linear recurrences, both defined by the relation u n = u n− 2 + u n− 3 for n ≥ 3. They differ in their initial conditions resulting in different sequences. The Padovan sequence begins with P 0 = P 1 = P 2 = 1, whereas the Perrin sequence starts with R 0 = 3, R 1 = 0 , and R 2 = 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 a 1 < a 2 < a 3 < a 4 such that all pairwise products a i a j + 1 (for i ≠ j ) belong to the Padovan or Perrin sequence, and we prove that the answer is negative.