Perfect powers in the summatory function of the power tower

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 (2010)

  • Volume: 22, Issue: 3, page 703-718
  • ISSN: 1246-7405

Abstract

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Let ( a n ) n ≥ 1 be the sequence given by a 1 = 1 and a n = n a n - 1 for n ≥ 2 . In this paper, we show that the only solution of the equation a 1 + ⋯ + a n = m l is in positive integers l > 1 , m and n is m = n = 1 .

How to cite

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Luca, Florian, and Marques, Diego. "Perfect powers in the summatory function of the power tower." Journal de Théorie des Nombres de Bordeaux 22.3 (2010): 703-718. <http://eudml.org/doc/116428>.

@article{Luca2010,
abstract = {Let $(a_n)_\{n\ge 1\}$ be the sequence given by $a_1=1$ and $a_n=n^\{a_\{n-1\}\}$ for $n\ge 2$. In this paper, we show that the only solution of the equation\[ a\_1+\cdots +a\_n=m^l \]is in positive integers $l&gt;1,~m$ and $n$ is $m=n=1$.},
affiliation = {Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México; Departamento de Matemática Universidade de Brasília Brasília, DF, Brazil},
author = {Luca, Florian, Marques, Diego},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {exponential Diophantine equations},
language = {eng},
number = {3},
pages = {703-718},
publisher = {Université Bordeaux 1},
title = {Perfect powers in the summatory function of the power tower},
url = {http://eudml.org/doc/116428},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Luca, Florian
AU - Marques, Diego
TI - Perfect powers in the summatory function of the power tower
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 3
SP - 703
EP - 718
AB - Let $(a_n)_{n\ge 1}$ be the sequence given by $a_1=1$ and $a_n=n^{a_{n-1}}$ for $n\ge 2$. In this paper, we show that the only solution of the equation\[ a_1+\cdots +a_n=m^l \]is in positive integers $l&gt;1,~m$ and $n$ is $m=n=1$.
LA - eng
KW - exponential Diophantine equations
UR - http://eudml.org/doc/116428
ER -

References

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  7. N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences. http://www.research.att.com/ njas/sequences/. Zbl1274.11001
  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. Zbl1113.11040MR1990621
  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. Zbl0838.11065MR1367580
  11. R. T. Worley, Estimating | α - p / q | . J. Austral. Math. Soc. Ser. A 31 (1981), 202–206. Zbl0465.10026MR629174
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