# $S$-integral solutions to a Weierstrass equation

Journal de théorie des nombres de Bordeaux (1997)

- Volume: 9, Issue: 2, page 281-301
- ISSN: 1246-7405

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topde Weger, Benjamin M. M.. "$S$-integral solutions to a Weierstrass equation." Journal de théorie des nombres de Bordeaux 9.2 (1997): 281-301. <http://eudml.org/doc/247994>.

@article{deWeger1997,

abstract = {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.},

author = {de Weger, Benjamin M. M.},

journal = {Journal de théorie des nombres de Bordeaux},

keywords = {elliptic curves; cubic diophantine equations; integral points; rational solutions; -unit equations; Weierstrass equation},

language = {eng},

number = {2},

pages = {281-301},

publisher = {Université Bordeaux I},

title = {$S$-integral solutions to a Weierstrass equation},

url = {http://eudml.org/doc/247994},

volume = {9},

year = {1997},

}

TY - JOUR

AU - de Weger, Benjamin M. M.

TI - $S$-integral solutions to a Weierstrass equation

JO - Journal de théorie des nombres de Bordeaux

PY - 1997

PB - Université Bordeaux I

VL - 9

IS - 2

SP - 281

EP - 301

AB - 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.

LA - eng

KW - elliptic curves; cubic diophantine equations; integral points; rational solutions; -unit equations; Weierstrass equation

UR - http://eudml.org/doc/247994

ER -

## References

top- [B] Yu. Bilu, "Solving superelliptic Diophantine equations by the method of Gelfond-Baker ", Preprint 94-09, Mathématiques Stochastiques, Univ. Bordeaux2 [1994].
- [BH] Yu. Bilu AND G. Hanrot, "Solving superelliptic Diophantine equations by Baker's method", Compos. Math., to appear. Zbl0915.11065
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- [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. Zbl0899.11012MR1446509
- [RU] G. Remond AND F. Urfels, "Approximation diophantienne de logarithmes elliptiques p-adiques", J. Number Th.57 [1996], 133-169. Zbl0853.11055MR1378579
- [S] N.P. Smart, "S-integral points on elliptic curves", Math. Proc. Cambridge Phil. Soc.116 [1994], 391-399. Zbl0817.11031MR1291748
- [ST] R.J. Stroeker AND N. Tzanakis, "Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms", Acta Arith.67 [1994], 177-196. Zbl0805.11026MR1291875
- [SW1] R.J. Stroeker AND B.M.M. De Weger, "On a quartic diophantine equation", Proc. Edinburgh Math. Soc.39 [1996], 97-115. Zbl0861.11020MR1375670
- [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. Zbl0858.11016MR1379397
- [TW1] N. Tzanakis AND B.M.M. De Weger, "On the practical solution of the Thue equation", J. Number Th.31 [1989], 99-132. Zbl0657.10014MR987566
- [TW2] N. Tzanakis AND B.M.M. De Weger, "How to explicitly solve a Thue-Mahler equation", Compos. Math.84 [1992], 223-288. Zbl0773.11023MR1189890
- [Y] K.R. Yu, "Linear forms in p-adic logarithms III", Compos. Math.91 [1994], 241-276. Zbl0819.11025MR1273651

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