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 $m = n = 1$ .

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},
     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},
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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/

References
Cited by

[1] Y. Bugeaud and M. Laurent, Minoration effective de la distance p-adique entre puissances de nombres algébriques. J. Number Theory 61 (1996), 31–42. | MR | Zbl

[2] Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers. Ann. of Math. (2) 163 (2006), 969–1018. | MR | Zbl

[3] L. K. Hua, Introduction to number theory. Springer-Verlag, 1982. | MR | Zbl

[4] E. Landau, Verallgemeinerung eines Polyaschen Satzes auf algebraische Zahlkörper. Nachr. Kgl. Ges. Wiss. Göttingen, Math.-Phys. Kl. (1918), 478–488.

[5] M. Laurent, M. Mignotte and Yu. Nesterenko, Formes linéaires en deux logarithmes et déterminants d’interpolation. J. Number Theory 55 (1995), 285–321. | MR | Zbl

[6] D. Poulakis, Solutions entières de l’équation $Y^{m} = f (X)$ . Sém. Théor. Nombres Bordeaux (2) 3 (1991), 187–199. | Numdam | MR | Zbl

[7] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences. http://www.research.att.com/ njas/sequences/.

[8] J. Sondow, An irrationality measure for Liouville numbers and conditional measures for Euler’s constant. Preprint, 2003, http://arXiv.org/abs/math.NT/0307308. | MR

[9] J. Sondow, Irrationality measures, irrationality bases, and a theorem of Jarnik. Preprint, 2004, http://arXiv.org/abs/math.NT/0406300.

[10] P. Voutier, An effective lower bound for the height of algebraic numbers. Acta Arith. 74 (1996), 81–95. | MR | Zbl

[11] R. T. Worley, Estimating $| α - p / q |$ . J. Austral. Math. Soc. Ser. A 31 (1981), 202–206. | MR | Zbl

[12] K. Yu, $p$ -adic logarithmic forms and group varieties I. J. Reine Angew. Math. 502 (1998), 29–92. | MR | Zbl

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