Inglês (pdf)
Artigo em XML
Referências do artigo
Tradução automática
Enviar este artigo por email
Permalink
Text complete and PDF
Serviços Personalizados
Journal
Artigo
Indicadores
-
Citado por SciELO -
Acessos
Links relacionados
-
Citado por Google -
Similares em
SciELO -
Similares em Google
Compartilhar
Revista Colombiana de Matemáticas
versão impressa ISSN 0034-7426
Rev.colomb.mat. vol.53 no.1 Bogotá jan./jun. 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.
References
[1] H. Davenport A. Baker, The equations 3X2 - 2 = Y2 and 8X2 - 7 = Z2, Quart. J. Math. Oxford 20 (1969), no. 2, 129-137. [ Links ]
[2] J. J. Bravo, C. A. Gómez, and F. Luca, Powers of two as sums of two k-Fibonacci numbers, Miskolc Math. Notes 17 (2016), no. 1, 85-100. [ Links ]
[3] J. J. Bravo, F. Luca, and K. Yazan, On Pillai's problem with Tribonacci numbers and Powers of 2, Bull. Korean Math. Soc. 54 (2017), no. 3, 1069-1080. [ Links ]
[4] Y. Bugeaud, M. Mignotte, and S. Siksek, Classical and modular approaches to exponential diophantine equations I: Fibonacci and Lucas perfect powers, Ann. of Math. 163 (2006), 969-1018. [ Links ]
[5] K. C. Chim, I. Pink, and V. Ziegler, On a variant of Pillai's problem, Int. J. Number Theory 7 (2017), 1711-1727. [ Links ]
[6] ______, On a variant of Pillai's problem II, J. Number Theory 183 (2018), 269-290. [ Links ]
[7] M. Ddamulira, C. A. Gómez, and F. Luca, On a problem of Pillai with k-generalized Fibonacci numbers and powers of 2, Monatsh. Math., 2018, https://doi.org/10.1007/s00605-018-1155-1. [ Links ]
[8] M. Ddamulira, F. Luca, and M. Rakotomalala, On a problem of Pillai with Fibonacci and powers of2, Proc. Indian Acad. Sci. (Math. Sci.) 127 (2017), no. 3, 411-421. [ Links ]
[9] A. Dujella and A. Petho, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford 49 (1998), no. 3, 291-306. [ Links ]
[10] S. Hernandez Hernández, F. Luca, and L. M. Rivera, On Pillai's problem with the Fibonacci and Pell sequences, Accepted in the Bol. Soc. Mat. Mexicana (2018). [ Links ]
[11] A. Herschfeld, The equation 2 x-3y = d, Bull. Amer. Math. Soc. 42 (1936), 231-234. [ Links ]
[12] E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms ofalgebraic numbers II, Izv. Math. 64 (2000), no. 6, 1217-1269. [ Links ]
[13] S. S. Pillai, On a x - by = c, J. Indian Math. Soc. 2 (1936), 119-122. [ Links ]
[14] ______, On the equation 2x - 3y = 2 X +3Y, Bull. Calcutta Math. Soc. 37 (1945), 15-20. [ Links ]
[15] S. Guzman Sanchez and F. Luca, Linear combinations of factorials and S-units in a binary recurrence sequence, Ann. Math. Quebec 38 (2014), 169-188. [ Links ]
[16] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, https://oeis.org/. [ Links ]
[17] R. J. Stroeker and R. Tijdeman, Diophantine equations, Computational methods in number theory, Math. Centre Tracts (155), Centre for Mathematics and Computer Science, Amsterdan (1982), 321-369. [ Links ]
Received: June 2018; Accepted: October 2018
This is an open-access article distributed under the terms of the Creative Commons Attribution License
