Perfect powers in the summatory function of the power tower

Florian Luca; Diego Marques

doi:10.5802/jtnb.740

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, Tome 22 (2010) no. 3, pp. 703-718

Abstract (VO)
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 $m = n = 1$ .

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DOI : 10.5802/jtnb.740

Affiliations des auteurs :

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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@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},
 year = {2010},
 publisher = {Universit\'e Bordeaux 1},
 volume = {22},
 number = {3},
 doi = {10.5802/jtnb.740},
 mrnumber = {2769339},
 zbl = {1231.11040},
 language = {en},
 url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.740/}
}

TY - JOUR
AU - Florian Luca
AU - Diego Marques
TI - Perfect powers in the summatory function of the power tower
JO - Journal de théorie des nombres de Bordeaux
PY - 2010
SP - 703
EP - 718
VL - 22
IS - 3
PB - Université Bordeaux 1
UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.740/
DO - 10.5802/jtnb.740
LA - en
ID - JTNB_2010__22_3_703_0
ER -

%0 Journal Article
%A Florian Luca
%A Diego Marques
%T Perfect powers in the summatory function of the power tower
%J Journal de théorie des nombres de Bordeaux
%D 2010
%P 703-718
%V 22
%N 3
%I Université Bordeaux 1
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.740/
%R 10.5802/jtnb.740
%G en
%F JTNB_2010__22_3_703_0

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

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