S-integral solutions to a Weierstrass equation
Benjamin M. M. de Weger
Journal de Théorie des Nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 281-301.

On détermine les solutions rationnelles de l’équation diophantienne y 2 =x 3 -228x+848 dont les dénominateurs sont des puissances de 2. On applique une idée de Yuri Bilu, qui évite le recours à des équations de Thue et de Thue-Mahler, et qui permet d’obtenir des équations aux (S-) unités à quatre termes dotées de propriétés spéciales, que l’on résout par la théorie des formes linéaires en logarithmes réels et p-adiques.

The rational solutions with as denominators powers of 2 to the elliptic diophantine equation y 2 =x 3 -228x+848 are determined. An idea of Yuri Bilu is applied, which avoids Thue and Thue-Mahler equations, and deduces four-term (S-) unit equations with special properties, that are solved by linear forms in real and p-adic logarithms.

@article{JTNB_1997__9_2_281_0,
     author = {de Weger, Benjamin M. M.},
     title = {$S$-integral solutions to a {Weierstrass} equation},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {281--301},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {2},
     year = {1997},
     doi = {10.5802/jtnb.203},
     zbl = {0898.11009},
     mrnumber = {1617399},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.203/}
}
Benjamin M. M. de Weger. $S$-integral solutions to a Weierstrass equation. Journal de Théorie des Nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 281-301. doi : 10.5802/jtnb.203. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.203/

[B] Yu. Bilu, "Solving superelliptic Diophantine equations by the method of Gelfond-Baker ", Preprint 94-09, Mathématiques Stochastiques, Univ. Bordeaux 2 [1994].

[BH] Yu. Bilu AND G. Hanrot, "Solving superelliptic Diophantine equations by Baker's method", Compos. Math., to appear. | Zbl 0915.11065

[BW] A. Baker AND G. Wüstholz, "Logarithmic forms and group varieties ", J. reine angew. Math. 442 [1993], 19-62. | MR 1234835 | Zbl 0788.11026

[D] S. David, Minorations de formes linéaires de logarithmes elliptiques, Mém. Soc. Math. de France, Num.62 [1995]. | Numdam | MR 1385175 | Zbl 0859.11048

[GPZ1] J. Gebel, A. Pethö AND H.G. Zimmer, "Computing integral points on elliptic curves", Acta Arith. 68 [1994], 171-192. | MR 1305199 | Zbl 0816.11019

[GPZ2] J. Gebel, A. Pethö AND H.G. Zimmer, "Computing S-integral points on elliptic curves", in: H. COHEN (ED.), Algorithmic Number Theory, Proceedings ANTS-II, Lecture Notes in Computer Science VOl. 1122, Springer-Verlag, Berlin [1996], pp. 157-171. | MR 1446509 | Zbl 0899.11012

[RU] G. Remond AND F. Urfels, "Approximation diophantienne de logarithmes elliptiques p-adiques", J. Number Th. 57 [1996], 133-169. | MR 1378579 | Zbl 0853.11055

[S] N.P. Smart, "S-integral points on elliptic curves", Math. Proc. Cambridge Phil. Soc. 116 [1994], 391-399. | MR 1291748 | Zbl 0817.11031

[ST] R.J. Stroeker AND N. Tzanakis, "Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms", Acta Arith. 67 [1994], 177-196. | MR 1291875 | Zbl 0805.11026

[SW1] R.J. Stroeker AND B.M.M. De Weger, "On a quartic diophantine equation", Proc. Edinburgh Math. Soc. 39 [1996], 97-115. | MR 1375670 | Zbl 0861.11020

[T] N. Tzanakis, "Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations", Acta Arith. 75 [1996], 165-190. | EuDML 206868 | MR 1379397 | Zbl 0858.11016

[TW1] N. Tzanakis AND B.M.M. De Weger, "On the practical solution of the Thue equation", J. Number Th. 31 [1989], 99-132. | MR 987566 | Zbl 0657.10014

[TW2] N. Tzanakis AND B.M.M. De Weger, "How to explicitly solve a Thue-Mahler equation", Compos. Math. 84 [1992], 223-288. | EuDML 90185 | Numdam | MR 1189890 | Zbl 0773.11023

[Y] K.R. Yu, "Linear forms in p-adic logarithms III", Compos. Math. 91 [1994], 241-276. | EuDML 90291 | Numdam | MR 1273651 | Zbl 0819.11025