Perfect powers among Fibonomial coefficients

Marques, Diego; Togbé, Alain

doi:10.1016/j.crma.2010.06.006

Number Theory

Perfect powers among Fibonomial coefficients
[Puissances parfaites parmi les coefficients Fibonomiaux]

Présenté par : Jean-Pierre Serre

Marques, Diego ¹ ; Togbé, Alain ²

¹ Departamento De Matemática, Universidade De Brasília, Brasília, DF, Brazil
² Mathematics Department, Purdue University North Central, 1401 S, U.S. 421, Westville, IN 46391, USA

Comptes Rendus. Mathématique, Tome 348 (2010) no. 13-14, pp. 717-720.

Résumé
Abstract

Soit $F_{n}$ le $n^{\overset{`}{e} me}$ nombre de Fibonacci. Pour $1 ⩽ k ⩽ m$ , soit

{[\begin{matrix} m \\ k \end{matrix}]}_{F} = \frac{F_{m} F_{m - 1} \dots F_{m - k + 1}}{F_{1} \dots F_{k}}

le coéfficient Fibonomial correspondant. En 2003, les puissances parfaites dans la suite de Fibonacci ont été complètement déterminées. Ainsi, les seules solutions de

F_{m} = y^{t}

, avec

m > 1

, sont

(m, y, t) = (6, 2, 3)

(12, 12, 2)

. Dans cet article, nous montrons que les seules solutions de l'équation diophantienne

{[\begin{matrix} m \\ k \end{matrix}]}_{F} = y^{t},

avec

m > k + 1

t > 1

, sont celles pour lesquelles

k = 1

, qui sont

(m, k, y, t) = (6, 1, 2, 3)

(12, 1, 12, 2)

Let $F_{n}$ be the nth Fibonacci number. For $1 ⩽ k ⩽ m$ , let

{[\begin{matrix} m \\ k \end{matrix}]}_{F} = \frac{F_{m} F_{m - 1} \dots F_{m - k + 1}}{F_{1} \dots F_{k}}

be the corresponding Fibonomial coefficient. In 2003, the problem of determining the perfect powers in the Fibonacci sequence was completely solved. In fact, the only solutions of

F_{m} = y^{t}

, with

m > 2

, are

(m, y, t) = (6, 2, 3)

(12, 12, 2)

. In this paper, we prove that the only solutions of the Diophantine equation

{[\begin{matrix} m \\ k \end{matrix}]}_{F} = y^{t},

with

m > k + 1

and

t > 1

, are those related to

k = 1

, that is

(m, k, y, t) = (6, 1, 2, 3)

and

(12, 1, 12, 2)

Reçu le : 2010-05-19
Accepté le : 2010-06-02
Publié le : 2010-06-20

DOI : 10.1016/j.crma.2010.06.006

Affiliations des auteurs :

Marques, Diego ¹ ; Togbé, Alain ²

¹ Departamento De Matemática, Universidade De Brasília, Brasília, DF, Brazil
² Mathematics Department, Purdue University North Central, 1401 S, U.S. 421, Westville, IN 46391, USA

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     title = {Perfect powers among {Fibonomial} coefficients},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {717--720},
     publisher = {Elsevier},
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Marques, Diego; Togbé, Alain. Perfect powers among Fibonomial coefficients. Comptes Rendus. Mathématique, Tome 348 (2010) no. 13-14, pp. 717-720. doi : 10.1016/j.crma.2010.06.006. http://www.numdam.org/articles/10.1016/j.crma.2010.06.006/

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