Two exponential diophantine equations

Dominik J. Leitner

doi:10.5802/jtnb.773

Dominik J. Leitner ¹

¹ Mathematisches Institut Universität Basel Rheinsprung 21 4051 Basel, Switzerland

Journal de théorie des nombres de Bordeaux, Tome 23 (2011) no. 2, pp. 479-487.

Abstract (VO)
Résumé

The equation $3^{a} + 5^{b} - 7^{c} = 1$ , to be solved in non-negative rational integers $a, b, c$ , has been mentioned by Masser as an example for which there is still no algorithm to solve completely. Despite this, we find here all the solutions. The equation $y^{2} = 3^{a} + 2^{b} + 1$ , to be solved in non-negative rational integers $a, b$ and a rational integer $y$ , has been mentioned by Corvaja and Zannier as an example for which the number of solutions is not yet known even to be finite. But we find here all the solutions too; there are in fact only six.

L’équation $3^{a} + 5^{b} - 7^{c} = 1$ , dont les inconnues $a, b, c$ sont des entiers positifs, a été mentionnée par Masser comme un exemple pour lequel il n’y a pas d’algorithme permettant une résolution complète. Malgré cela, nous trouvons ici toutes les solutions. L’équation $y^{2} = 3^{a} + 2^{b} + 1$ , dont les inconnues $a, b$ sont des entiers positifs et $y$ est un entier, a été mentionnée par Corvaja et Zannier comme un exemple dont on ignore si le nombre de solutions est fini. Mais nous trouvons également ici toutes les solutions ; il n’y en a en fait que six.

MR

DOI : 10.5802/jtnb.773

Affiliations des auteurs :

Dominik J. Leitner ¹

¹ Mathematisches Institut Universität Basel Rheinsprung 21 4051 Basel, Switzerland

@article{JTNB_2011__23_2_479_0,
     author = {Dominik J. Leitner},
     title = {Two exponential diophantine equations},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {479--487},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {23},
     number = {2},
     year = {2011},
     doi = {10.5802/jtnb.773},
     mrnumber = {2817941},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.773/}
}

TY  - JOUR
AU  - Dominik J. Leitner
TI  - Two exponential diophantine equations
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2011
SP  - 479
EP  - 487
VL  - 23
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.773/
DO  - 10.5802/jtnb.773
LA  - en
ID  - JTNB_2011__23_2_479_0
ER  -

%0 Journal Article
%A Dominik J. Leitner
%T Two exponential diophantine equations
%J Journal de théorie des nombres de Bordeaux
%D 2011
%P 479-487
%V 23
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.773/
%R 10.5802/jtnb.773
%G en
%F JTNB_2011__23_2_479_0

Dominik J. Leitner. Two exponential diophantine equations. Journal de théorie des nombres de Bordeaux, Tome 23 (2011) no. 2, pp. 479-487. doi : 10.5802/jtnb.773. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.773/

Bibliographie
Cité par

[AV] F. Amoroso and E. Viada, Small points on subvarieties of a torus. Duke Mathematical Journal 150(3) (2009), 407–442. | MR

[BF] J. L. Brenner and L. L. Foster, Exponential diophantine equations. Pacific Journal of Mathematics 101 (1982), 263–301. | MR | Zbl

[C] P. Corvaja, Problems and results on integral points on rational surfaces. In Diophantine geometry (ed. U. Zannier), Edizioni della Normale 2007, 123–141. | MR | Zbl

[CZ1] P. Corvaja and U. Zannier, On the diophantine equation $f (a^{m}, y) = b^{n}$ . Acta Arithmetica 94 (2000), 25–40. | MR | Zbl

[CZ2] P. Corvaja and U. Zannier, Some cases of Vojta’s conjecture on integral points over function fields. Journal of Algebraic Geometry 17 (2008), 295–333. | MR

[CZ3] P. Corvaja and U. Zannier, Applications of the subspace theorem to certain diophantine problems. In Diophantine approximation, Developments in Mathematics 18 (eds H.P. Schlickewei, K. Schmidt, R.F. Tichy), Springer 2008, 161–174. | MR

[ESS] J.-H. Evertse, H. P. Schlickewei and W. M. Schmidt, Linear equations in variables which lie in a multiplicative group. Annals of Mathematics 155 (2002), 807–836. | MR | Zbl

[HS] M. Hindry and J. H. Silverman, Diophantine Geometry. Springer, 2000. | MR | Zbl

[L] S. Lang, Fundamentals of Diophantine Geometry. Springer, 1983. | MR | Zbl

[M] D. W. Masser, Mixing and linear equations over groups in positive characteristic. Israel Journal of Mathematics 142 (2004), 189–204. | MR | Zbl

[V] P. Vojta, A more general $a b c$ conjecture. International Mathematics Research Notices 21 (1998), 1103–1116. | MR | Zbl

[Z1] U. Zannier, Some applications of diophantine approximation to diophantine equations. Forum Editrice Universitaria Udinese S.r.l. (2003).

[Z2] U. Zannier, Polynomial squares of the form $a X^{m} + b {(1 - X)}^{n} + c$ . Rendiconti del Seminario Matematico della Università di Padova 112 (2004), 1–9. | Numdam | MR | Zbl

[Z3] U. Zannier, Diophantine equations with linear recurrences. An overview of some recent progress. Journal de Théorie des Nombres de Bordeaux 17 (2005), 432–435. | Numdam | MR | Zbl

Cité par Sources :