Two exponential diophantine equations
Journal de Théorie des Nombres de Bordeaux, Tome 23 (2011) no. 2, pp. 479-487.

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.

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.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.773
@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},
     zbl = {pre05955277},
     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
DA  - 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/
UR  - https://www.ams.org/mathscinet-getitem?mr=2817941
UR  - https://zbmath.org/?q=an%3Apre05955277
UR  - https://doi.org/10.5802/jtnb.773
DO  - 10.5802/jtnb.773
LA  - en
ID  - JTNB_2011__23_2_479_0
ER  - 
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/

[AV] F. Amoroso and E. Viada, Small points on subvarieties of a torus. Duke Mathematical Journal 150(3) (2009), 407–442. | MR 2582101 | Zbl pre05688238

[BF] J. L. Brenner and L. L. Foster, Exponential diophantine equations. Pacific Journal of Mathematics 101 (1982), 263–301. | MR 675401 | Zbl 0447.10021

[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 2349651 | Zbl 1144.11049

[CZ1] P. Corvaja and U. Zannier, On the diophantine equation f(a m ,y)=b n . Acta Arithmetica 94 (2000), 25–40. | MR 1762454 | Zbl 0963.11020

[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 2369088 | Zbl pre05270801

[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 2487692 | Zbl pre05354079

[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 1923966 | Zbl 1026.11038

[HS] M. Hindry and J. H. Silverman, Diophantine Geometry. Springer, 2000. | MR 1745599 | Zbl 0948.11023

[L] S. Lang, Fundamentals of Diophantine Geometry. Springer, 1983. | MR 715605 | Zbl 0528.14013

[M] D. W. Masser, Mixing and linear equations over groups in positive characteristic. Israel Journal of Mathematics 142 (2004), 189–204. | MR 2085715 | Zbl 1055.37009

[V] P. Vojta, A more general abc conjecture. International Mathematics Research Notices 21 (1998), 1103–1116. | MR 1663215 | Zbl 0923.11059

[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 aX m +b(1-X) n +c. Rendiconti del Seminario Matematico della Università di Padova 112 (2004), 1–9. | Numdam | MR 2109949 | Zbl 1167.11304

[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 2152233 | Zbl 1162.11330

Cité par Sources :