×

zbMATH — the first resource for mathematics

\(S\)-integral solutions to a Weierstrass equation. (English) Zbl 0898.11009
The author determines the rational solutions (with a power of \(2\) in the denominator) to the equation \[ y^2 = x^3 - 228 x + 848. \] This is an elliptic curve of rank two. There are a variety of standard techniques to solve this problem: For instance using a reduction to Thue-Mahler equations or using elliptic logarithms (and their \(p\)-adic generalizations). However, in this paper the author takes the novel approach of reduction to four term \(S\)-unit equations of a special form. The author attributes this idea to Y. Bilu.
The approach in the current paper has a number of advantages. One does not need unproved estimates for linear forms in \(p\)-adic elliptic logarithms, nor does one need to reduce to a set of Thue-Mahler equations, which can lead to huge computational problems. In addition, the paper contains a detailed explanation of the calculations performed to solve the equation.

MSC:
11D25 Cubic and quartic Diophantine equations
11G05 Elliptic curves over global fields
14G05 Rational points
PDF BibTeX XML Cite
Full Text: DOI EMIS Numdam EuDML
References:
[1] Bilu, Yu., “Solving superelliptic Diophantine equations by the method of Gelfond-Baker ”, Preprint 94-09, , Univ. Bordeaux2 [1994].
[2] Bilu, Yu. AND Hanrot, G., “Solving superelliptic Diophantine equations by Baker”s method”, Compos. Math., to appear. · Zbl 0915.11065
[3] Baker, A. AND Wüstholz, G., “Logarithmic forms and group varieties ”, J. reine angew. Math.442 [1993], 19-62. · Zbl 0788.11026
[4] David, S., Minorations de formes linéaires de logarithmes elliptiques, Mém. Soc. Math. de France, Num.62 [1995]. · Zbl 0859.11048
[5] Gebel, J., Pethö, A. AND Zimmer, H.G., “Computing integral points on elliptic curves”, Acta Arith.68 [1994], 171-192. · Zbl 0816.11019
[6] Gebel, J., Pethö, A. AND Zimmer, H.G., “Computing S-integral points on elliptic curves”, in: H. COHEN (ED.), Algorithmic Number Theory, Proceedings ANTS-II, VOl. 1122, Springer-Verlag, Berlin [1996], pp. 157-171. · Zbl 0899.11012
[7] Remond, G. AND Urfels, F., “Approximation diophantienne de logarithmes elliptiques p-adiques”, J. Number Th.57 [1996], 133-169. · Zbl 0853.11055
[8] Smart, N.P., “S-integral points on elliptic curves”, Math. Proc. Cambridge Phil. Soc.116 [1994], 391-399. · Zbl 0817.11031
[9] Stroeker, R.J. AND Tzanakis, N., “Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms”, Acta Arith.67 [1994], 177-196. · Zbl 0805.11026
[10] Stroeker, R.J. AND De Weger, B.M.M., “On a quartic diophantine equation”, Proc. Edinburgh Math. Soc.39 [1996], 97-115. · Zbl 0861.11020
[11] Tzanakis, N., “Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations”, Acta Arith.75 [1996], 165-190. · Zbl 0858.11016
[12] Tzanakis, N. AND De Weger, B.M.M., “On the practical solution of the Thue equation”, J. Number Th.31 [1989], 99-132. · Zbl 0657.10014
[13] Tzanakis, N. AND De Weger, B.M.M., “How to explicitly solve a Thue-Mahler equation”, Compos. Math.84 [1992], 223-288. · Zbl 0773.11023
[14] Yu, K.R., “Linear forms in p-adic logarithms III”, Compos. Math.91 [1994], 241-276. · Zbl 0819.11025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.