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Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
Rev.colomb.mat. vol.53 no.1 Bogotá Jan./June 2019
Original articles
Pillai's problem with Padovan numbers and powers of two
El problema de Pillai con números de Padovan y potencias de dos
1 Universidad Autónoma de Zacatecas, Zacatecas, México. Unidad Académica de Matemáticas Universidad Autónoma de Zacatecas, Campus II, Calzada Solidaridad entronque Paseo a la Bufa C.P. 98000 Zacatecas, Zac. México. e-mail: aceciliagarcia.lomeli@gmail.com
2 Universidad Autónoma de Zacatecas, Zacatecas, México. Unidad Académica de Matemáticas Universidad Autónoma de Zacatecas, Campus II, Calzada Solidaridad entronque Paseo a la Bufa C.P. 98000 Zacatecas, Zac. México. e-mail: shh@uaz.edu.mx
Let (P n ) n≥0 be the Padovan sequence given by P 0 = 0, P 1 = P 2 = 1 and the recurrence formula P n+3 = P n+1 + P n for all n ≥ 0. In this note we study and completely solve the Diophantine equation P n - 2 m = P n1 -2 m1 in non-negative integers (n,m,n 1 ,m 1 ).
Key words and phrases: Padovan sequence; Pillai's problem; linear forms in logarithms; reduction method
Sea (P n ) n≥0 la sucesión de Padovan dada mediante P 0 = 0, P 1 = P 2 = 1 y la fórmula de recurrencia P n+3 = P n+1 + P n para todo n ≥ 0. En esta nota estudiamos y resolvemos completamente la ecuación diofántica P n - 2 m = P n1 -2 m1 en enteros no negativos (n,m,n 1 ,m 1 ).
Palabras y frases clave: Sucesión de Padovan; Problema de Pillai; Formas lineales en logaritmos; método de reducción
Acknowledgements
We would like to thank the anonymous referee for painstaking reading, whose valuable suggestions improve the presentation of this work. The first author was supported by a CONACyT Doctoral Fellowship and partly supported by Fundación Kovalevskaia de la Sociedad Matemática Mexicana. We thank F. Luca for very valuable comments and suggestions. The second author thanks Leticia A. Ramírez for kind and generous support and Lidia Gonzalez García for valuable bibliography support. He also thanks L.M. Rivera for a tutorial on Mathematica.
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Received: June 2018; Accepted: October 2018
This is an open-access article distributed under the terms of the Creative Commons Attribution License
