Perfect powers in the summatory function of the power tower

Luca, Florian; Marques, Diego

doi:10.5802/jtnb.740

Luca, Florian ¹; Marques, Diego ²

¹ Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México
² Departamento de Matemática Universidade de Brasília Brasília, DF, Brazil

Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 3, pp. 703-718.

Abstract
Résumé

Let ${(a_{n})}_{n \geq 1}$ be the sequence given by $a_{1} = 1$ and $a_{n} = n^{a_{n - 1}}$ for $n \geq 2$ . In this paper, we show that the only solution of the equation

a_{1} + \dots + a_{n} = m^{l}

is in positive integers $l > 1, m$ and $n$ is $m = n = 1$ .

Soit ${(a_{n})}_{n \geq 1}$ la suite donnée par $a_{1} = 1$ et $a_{n} = n^{a_{n - 1}}$ pour $n \geq 2$ . Dans cet article, on montre que la seule solution de l’équation

a_{1} + \dots + a_{n} = m^{l}

avec des entiers positifs $l > 1, m$ et $n$ est $m = n = 1$ .

MR Zbl

DOI: 10.5802/jtnb.740

Author's affiliations:

Luca, Florian ¹; Marques, Diego ²

¹ Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México
² Departamento de Matemática Universidade de Brasília Brasília, DF, Brazil

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     author = {Luca, Florian and Marques, Diego},
     title = {Perfect powers in the summatory function of the power tower},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {703--718},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {3},
     year = {2010},
     doi = {10.5802/jtnb.740},
     zbl = {1231.11040},
     mrnumber = {2769339},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.740/}
}

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Luca, Florian; Marques, Diego. Perfect powers in the summatory function of the power tower. Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 3, pp. 703-718. doi : 10.5802/jtnb.740. http://archive.numdam.org/articles/10.5802/jtnb.740/

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