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Solving elliptic Diophantine equations: the general cubic case. (English) Zbl 0930.11015
The Elliptic Logarithm Method for computing explicitly all integral solutions of a diophantine equation that defines an elliptic curve over \(\mathbb Q\) (or, more generally, over a number field) as a practical method, has been developed by R. J. Stroeker and N. Tzanakis [Acta Arith. 67, 177–196 (1994; Zbl 0805.11026)] and independently by J. Gebel, A. Pethő and H. G. Zimmer [Acta Arith. 68, 171–192 (1994; Zbl 0816.11019)]. In those papers the general Weierstrass equation is studied.
Later, modifications of this method made possible the solution of other classes of elliptic diophantine equations: quartic polynomial in one variable \(=\square\) [N. Tzanakis, Acta Arith. 75, 165–190 (1996; Zbl 0858.11016)]; cubic Weierstrass equations over number fields [N. P. Smart and N. M. Stephens, Math. Proc. Camb. Philos. Soc. 122, 9–16 (1997; Zbl 0881.11054)]; \(S\)-integral points on cubic Weierstrass equations [A. Pethő, H. G. Zimmer, J. Gebel and E. Hermann, Math. Proc. Camb. Philos. Soc. 127, 383–402 (1999; Zbl 0949.11033)]; Elliptic binomial Diophantine equations [seethe (forthcoming) paper with this title by the present authors, Math. Comput. 68, No. 227, 1257–1281 (1999; Zbl 0920.11014)].
In this paper the authors generalize the method of their last mentioned paper and develop a variation of the Elliptic Logarithm Method for finding explicitly all (rational) integral points on any given cubic elliptic equation. The paper’s style is clear, vivid and friendly. As examples, the following equations are discussed, the third and the last being solved in detail: \[ {n\choose 3}={m\choose 6},\quad u(u-1)(u+1)={m\choose 3},\quad u(u-1)(u+1)={m\choose 6},\quad {n\choose k}={n-1\choose k+2}, \]
\[ u^3-u=48v^3-192v^2+144v,\quad 720u^3-2880u^2+2160u=v^3-4v^2+3v, \]
\[ -15u^3+45u^2v-15uv^2+v^3+90u^2-210uv+40v^2-120u+184v=0, \]
\[ -105u^3+105u^2v-21uv^2+v^3+630u^2-462uv+52v^2-840u+360v=0. \]

MSC:
11D25 Cubic and quartic Diophantine equations
11G05 Elliptic curves over global fields
11Y50 Computer solution of Diophantine equations
11J86 Linear forms in logarithms; Baker’s method
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