Perfect powers among Fibonomial coefficients

Diego Marques; Alain Togbé

doi:10.1016/j.crma.2010.06.006

Number Theory

Perfect powers among Fibonomial coefficients

Presented by: Jean-Pierre Serre

Diego Marques ¹; Alain Togbé 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, Volume 348 (2010) no. 13-14, pp. 717-720.

Abstract
Résumé

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)

.

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

et

t > 1

, sont celles pour lesquelles

k = 1

, qui sont

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

et

(12, 1, 12, 2)

.

Received: 2010-05-19
Accepted: 2010-06-02
Published online: 2010-06-20

DOI: 10.1016/j.crma.2010.06.006

Author's affiliations:

Diego Marques ¹; Alain Togbé 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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     author = {Diego Marques and Alain Togb\'e},
     title = {Perfect powers among {Fibonomial} coefficients},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {717--720},
     publisher = {Elsevier},
     volume = {348},
     number = {13-14},
     year = {2010},
     doi = {10.1016/j.crma.2010.06.006},
     language = {en},
}

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Diego Marques; Alain Togbé. Perfect powers among Fibonomial coefficients. Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 717-720. doi : 10.1016/j.crma.2010.06.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.06.006/

References
Cited by

[1] M. Abouzaid Les nombres de Lucas et Lehmer sans diviseur primitif, J. Théor. Nombres Bordeaux, Volume 18 (2006), pp. 299-313

[2] Yu. Bilu; G. Hanrot; P. Voutier Existence of primitive divisors of Lucas and Lehmer numbers (with an appendix by M. Mignotte), J. Reine Angew. Math., Volume 539 (2001), pp. 75-122

[3] Y. Bugeaud; M. Mignotte; S. Siksek Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas powers, Ann. of Math., Volume 163 (2006), pp. 969-1018

[4] F. Luca; T.N. Shorey Diophantine equations with products of consecutive terms in Lucas sequences, J. Number Theory, Volume 114 (2005), pp. 298-311

[5] P. Ribenboim My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, New York, 2000

[6] J.J. Sylvester On arithmetical series, Messenger Math., Volume 21 (1892), pp. 1-19 (87–120)

[7] M. Ward The prime divisors of Fibonacci numbers, Pacific J. Math., Volume 11 (1961) no. 1, pp. 379-386

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