S -integral solutions to a Weierstrass equation

Benjamin M. M. de Weger

Journal de théorie des nombres de Bordeaux (1997)

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

Abstract

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The rational solutions with as denominators powers of 2 to the elliptic diophantine equation y 2 = x 3 - 228 x + 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.

How to cite

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de 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

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  1. [B] Yu. Bilu, "Solving superelliptic Diophantine equations by the method of Gelfond-Baker ", Preprint 94-09, Mathématiques Stochastiques, Univ. Bordeaux2 [1994]. 
  2. [BH] Yu. Bilu AND G. Hanrot, "Solving superelliptic Diophantine equations by Baker's method", Compos. Math., to appear. Zbl0915.11065
  3. [BW] A. Baker AND G. Wüstholz, "Logarithmic forms and group varieties ", J. reine angew. Math.442 [1993], 19-62. Zbl0788.11026MR1234835
  4. [D] S. David, Minorations de formes linéaires de logarithmes elliptiques, Mém. Soc. Math. de France, Num.62 [1995]. Zbl0859.11048MR1385175
  5. [GPZ1] J. Gebel, A. Pethö AND H.G. Zimmer, "Computing integral points on elliptic curves", Acta Arith.68 [1994], 171-192. Zbl0816.11019MR1305199
  6. [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
  7. [RU] G. Remond AND F. Urfels, "Approximation diophantienne de logarithmes elliptiques p-adiques", J. Number Th.57 [1996], 133-169. Zbl0853.11055MR1378579
  8. [S] N.P. Smart, "S-integral points on elliptic curves", Math. Proc. Cambridge Phil. Soc.116 [1994], 391-399. Zbl0817.11031MR1291748
  9. [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
  10. [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
  11. [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
  12. [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
  13. [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
  14. [Y] K.R. Yu, "Linear forms in p-adic logarithms III", Compos. Math.91 [1994], 241-276. Zbl0819.11025MR1273651

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