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http://dx.doi.org/10.4134/BKMS.b160486

ON PILLAI'S PROBLEM WITH TRIBONACCI NUMBERS AND POWERS OF 2  

Bravo, Jhon J. (Departamento de Matematicas Universidad del Cauca)
Luca, Florian (School of Mathematics University of the Witwatersrand)
Yazan, Karina (Departamento de Matematicas Universidad del Cauca)
Publication Information
Bulletin of the Korean Mathematical Society / v.54, no.3, 2017 , pp. 1069-1080 More about this Journal
Abstract
The Tribonacci sequence ${\{T_n}\}_{n{\geq}0}$ resembles the Fibonacci sequence in that it starts with the values 0, 1, 1, and each term afterwards is the sum of the preceding three terms. In this paper, we find all integers c having at least two representations as a difference between a Tribonacci number and a power of 2. This paper continues the previous work [5].
Keywords
Tribonacci numbers; linear forms in logarithms; reduction method;
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1 G. P. Dresden and Z. Du, A simplified Binet formula for k-generalized Fibonacci numbers, J. Integer Seq. 17 (2014), no. 4, Article 14.4.7, 9 pp.
2 A. Dujella and A. Petho, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 195, 291-306.   DOI
3 A. Herschfeld, The equation $2^x$ − $3^y$ = d, Bull. Amer. Math. Soc. 41 (1935), 631.
4 A. Herschfeld, The equation $2^x$ − $3^y$ = d, Bull. Amer. Math. Soc. 42 (1936), no. 4, 231-234.   DOI
5 E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), no. 6, 125-180; translation in Izv. Math. 64 (2000), no. 6, 1217-1269.   DOI
6 P. Mihailescu, Primary cyclotomic units and a proof of Catalan's conjecture, J. Reine Angew. Math. 572 (2004), 167-195.
7 D. Z. Mo and R. Tijdeman, Exponential Diophantine equations with four terms, Indag. Math. (N.S.) 3 (1992), no. 1, 47-57.   DOI
8 S. S. Pillai, On $a^x$ − $b^y$ = c, J. Indian math. Soc. (N.S.) 2 (1936), 119-122.
9 S. S. Pillai, A correction to the paper "On $a^x$ − $b^y$ = c", J. Indian math. Soc. 2 (1937), 215.
10 C. L. Siegel, Uber einige Anwendungen diophantischer Approximationen, Abhandlungen Akad. Berlin (1929), no. 1, 70 S.
11 R. J. Stroeker and R. Tijdeman, Diophantine equations, in Computational Methods in Number Theory, Part II, , 321-369, vol. 155 of Math. Centre Tracts, Math. Centrum, Amsterdam, 1982.
12 A. Baker and H. Davenport, The equations $3x^2$ − 2 = $y^2$ and $8x^2$ − 7 = $z^2$, Quart. J Math. Ser. (2) 20 (1969), 129-137.   DOI
13 J. J. Bravo and F. Luca, On a conjecture about repdigits in k-generalized Fibonacci sequences, Publ. Math. Debrecen 82 (2013), no. 3-4, 623-639.   DOI
14 Y. Bugeaud, M.Maurice, and S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. of Math. 163 (2006), no. 3, 969-1018.   DOI
15 K. C. Chim, I. Pink, and V. Ziegler, On a variant of Pillai's problem, arXiv 1604.04719.
16 M. Ddamulira, F. Luca, and M. Rakotomalala, On a problem of Pillai with Fibonacci numbers and powers of 2, Preprint 2015.