Perfect powers in the summatory function of the power tower

Florian Luca; Diego Marques

doi:10.5802/jtnb.740

Florian Luca ; Diego Marques

Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 703-718.

Résumé
Abstract

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 $m = n = 1$ .

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 $m = n = 1$ .

Reçu le : 2009-08-06
Publié le : 2011-01-24

MR 2769339 | Zbl 1231.11040

DOI : https://doi.org/10.5802/jtnb.740

@article{JTNB_2010__22_3_703_0,
     author = {Florian Luca and Diego Marques},
     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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.740/}
}

Florian Luca; Diego Marques. Perfect powers in the summatory function of the power tower. Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 703-718. doi : 10.5802/jtnb.740. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.740/

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