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Pillai’s Problem with Padovan Numbers and Prime Powers
El problema de Pillai con números de Padovan y potencias de primos
DOI:
https://doi.org/10.15446/recolma.v59n1.122855Palabras clave:
Padovan numbers, Linear form in logarithms, Reduction method (en)Números de Padovan, Formas lineales en logaritmos, método de reducción (es)
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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.
Consideramos la succession de Padovan {Pn}n≥0, definida inductivamente por P0 = 0, P1 = P2 = 1, y la relación de recurrencia Pn+3 = Pn+1 + Pn para n ≥ 0. Este artículo utiliza el método de Baker-Davenport. Probamos que la ecuación Diofantina Pn − pm = Pn1 − pm1 admite sólo finitas soluciones enteras positivas n, m, n1, m1, donde p es un número primo fijo ≥ 5. Más aún, una vez fijo el valor de p, las soluciones pueden ser listadas de manera explícita. Mostramos este proceso para el caso p = 5.
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