On elliptic diophantine equations that defy Thue’s method: The case of the Ochoa curve. (English) Zbl 0824.11012

Stroeker, Roel J.; de Weger, Benjamin M. M.

Exp. Math. 3, No. 3, 209-220 (1994).

This is a very nice paper written by two experts in solving diophantine equations. They discuss the problem of finding all integer solutions to the equation \[ 3Y^ 2= 2X^ 3+ 385 X^ 2+ 256 X- 58195 \tag \(*\) \] proposed by Mr. Ochoa (but not used) at the 28-th International Mathematical Olympiad. The usual method of solution of such equations is to reduce them to a (usually very small) number of quartic Thue equations.
Nowadays, being equipped with Baker’s theory, the LLL-basis reduction algorithm, the powerful computers (even PC’s) and nice algebraic number theory packages (such as PARI), we feel quite certain that we can solve Thue equations, in particular of degree 4. A general method is described in [N. Tzanakis and the second author, On the practical solution of the Thue equation, J. Number Theory 31, 99-132 (1989; Zbl 0657.10014)]. However, if a number of “bad things”, related to the number fields appearing in the process implied by this method, do happen, then the whole task breaks down. This is what happens with \((*)\) [cf. R. K. Guy, The Ochoa curve, Crux Math. 16, 65-69 (1990)]. The authors, in a very lively way, explain their fruitless efforts when following this “Thue method”.
They present quite a number of fancy, huge algebraic numbers with which they came across (this gives a slightly humorous style in that part of the paper), thus convincing us that they did very well to give up with this method.
Then, they turn to a recent method, described in [the first author and N. Tzanakis, Acta Arith. 67, 177-196 (1994; Zbl 0805.11026)] (see the Introduction of that paper for some history and the underlying theory of this method). This consists in viewing (1) as the defining equation of an elliptic curve and finding all integral points on it by using the arsenal of elliptic curves plus a recent explicit lower bound for linear forms in elliptic logarithms due to S. David (but which has its origins in the work of N. Hirata-Kohno). Following this “elliptic curve” method, which is – at least conceptually – much more natural, the authors attempt anew the solution of \((*)\). Their efforts are crowned with success!
Reviewer’s recommendation to everybody interested in diophantine equations: Enjoy the reading of this nice paper!

Reviewer: N.Tzanakis (Iraklion)

MSC:

11D25	Cubic and quartic Diophantine equations
11Y50	Computer solution of Diophantine equations
11G05	Elliptic curves over global fields

Keywords:

elliptic diophantine equations; Ochoa curve; elliptic curve method; Thue equations; linear forms in elliptic logarithms

Citations:

Zbl 0657.10014; Zbl 0805.11026

Software:

APECS; ecdata; PARI/GP

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