Perfect powers in the summatory function of the power tower

Florian Luca; Diego Marques

doi:10.5802/jtnb.740

Florian Luca ¹; Diego Marques 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 $m = n = 1$ .

Received: 2009-08-06
Published online: 2011-01-24

MR: 2769339 | Zbl: 1231.11040

DOI: 10.5802/jtnb.740
Author's affiliations:

Florian Luca ¹; Diego Marques 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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     title = {Perfect powers in the summatory function of the power tower},
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     pages = {703--718},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
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Florian Luca; Diego Marques. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.740/

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