Números Tribonacci, S-unidades
y triplas diofánticas

CARLOS ALEXIS GÓMEZ RUIZ^*

Universidad del Valle, Departamento de Matemáticas, Cali, Colombia.

Resumen. La sucesión Tribonacci T := {T_n}_n ≥0 tiene valores iniciales T₀ = T₁ = 0, T₂ = 1 y cada término posterior es la suma de los tres términos precedentes. En este artículo, estudiamos la ecuación T_n = kT_m, donde k es una S-unidad, para un conjunto finito S de primos. Particularmente, mostramos que cualquier par de miembros de la tripla diofántica {9, 56, 103} asociada a T + 1, no se puede extender a otra tripla diofántica asociada a T + 1.

Palabras clave: Números Tribonacci, triplas diofánticas, formas lineales en logaritmos de números algebraicos.
MSC2010: 11B37, 11B39, 11J86.

Tribonacci numbers, S-units and
diophantine triples

Abstract. The Tribonacci sequence T := {T_n}_n ≥0 has initial values T₀ = T₁ = 0, T₂ = 1 and each term afterwards is the sum of the preceding three terms. In this paper, we study the equation T_n = kT_m, where k is an S-unit, for a finite set S of primes. In particular, we show that any two members of the diophantine triple {9, 56, 103} associated to T + 1, can not be extended to other diophantine triple associated to T + 1.

Keywords: Tribonacci numbers, diophantine triples, linear forms in logarithms of algebraic numbers.

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^*Email: carlos.a.gomez@correounivalle.edu.co
Recibido: 01 de julio de 2015, Aceptado: 04 de agosto de 2015.
Para citar este artículo: C.A. Gómez Ruiz, Números Tribonacci, S-unidades y triplas diofánticas, Rev. Integr. Temas Mat. 33 (2015), No. 2, 121-133.