El problema de Pillai con números de Padovan y potencias de primos

We consider the Padovan sequence {Pn}n≥0, defined by P0 = 0, P1 = P2 = 1, with subsequent terms given by the recurrence relation Pn+3 = Pn+1 + Pn for all n ≥ 0. In this paper, we use the methods of Baker-Davenport. We demonstrate that the Diophantine equation Pn − pm = Pn1 − pm1 admits only finitely many non-negative integer solutions n, m, n1, m1, where p is a fixed prime number ≥ 5. Additionally, once the value of p is specified, these solutions can be obtained explicitly. We address the case where p = 5.