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.

Soit (a n ) n1 la suite donnée par a 1 =1 et a n =n a n-1 pour n2. Dans cet article, on montre que la seule solution de l’équation

a 1 ++a n =m l

avec des entiers positifs l>1,m et n est m=n=1.

Let (a n ) n1 be the sequence given by a 1 =1 and a n =n a n-1 for n2. 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.

DOI : 10.5802/jtnb.740
Luca, Florian 1 ; Marques, Diego 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
@article{JTNB_2010__22_3_703_0,
     author = {Luca, Florian and Marques, Diego},
     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 = {http://www.numdam.org/articles/10.5802/jtnb.740/}
}
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
SP  - 703
EP  - 718
VL  - 22
IS  - 3
PB  - Université Bordeaux 1
UR  - http://www.numdam.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 Luca, Florian
%A Marques, Diego
%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 http://www.numdam.org/articles/10.5802/jtnb.740/
%R 10.5802/jtnb.740
%G en
%F JTNB_2010__22_3_703_0
Luca, Florian; Marques, Diego. 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. http://www.numdam.org/articles/10.5802/jtnb.740/

[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

Cité par Sources :