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Computing $$S$$-integral points on elliptic curves. (English) Zbl 0899.11012
Cohen, Henri (ed.), Algorithmic number theory. Second international symposium, ANTS-II, Talence, France, May 18-23, 1996. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1122, 157-171 (1996).
In a previous paper [Acta Arith. 68, 171-192 (1994; Zbl 0816.11019)] J. Gebel, A. Pethő and H. G. Zimmer developed a method, due to Lang and Zagier, for computing all the integral points on an elliptic curve $$E$$ over the field $$\mathbb{Q}$$ of rational numbers. This method requires knowledge of the Mordell-Weil group $$E(\mathbb{Q})$$ and relies on a method of S. David [Mém. Soc. Math. Fr., Nouv. Sér. 62 (1995; Zbl 0859.11048)] for computing lower bounds for linear forms in elliptic logarithms. In the present paper the authors develop a $$p$$-adic method for finding $$S$$-integral points on elliptic curves and list some tables concerning $$S$$-integral points on some curves of the form $$y^2= x^3+k$$. Their method relies on a new explicit bound for linear forms in two $$p$$-adic elliptic logarithms which was recently established by G. Rémond and F. Urfels [J. Number Theory 57, 133-169 (1996; Zbl 0853.11055)].
For the entire collection see [Zbl 0852.00023].

##### MSC:
 11D25 Cubic and quartic Diophantine equations 11G05 Elliptic curves over global fields 14H52 Elliptic curves 11-04 Software, source code, etc. for problems pertaining to number theory 11J86 Linear forms in logarithms; Baker’s method