Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations.

*(English)*Zbl 0858.11016In a joint paper [Acta Arith. 67, 177-196 (1994; Zbl 0805.11026)] the author and the reviewer gave a detailed description of a general method for the effective computation of all integer solutions of a given Weierstraß equation. This method, which relies on the theory of linear forms in elliptic logarithms, can be applied to any Weierstraß equation, provided the rank and a Mordell-Weil basis of the associated elliptic curve are explicitly known. Apart from the possible computational obstruction caused by this requirement, computational problems may also arise for curves of high rank at the final stages of the computation when a brute force search is usually needed. Apart from these difficulties, the method works rather well.

In the present paper the author extends the aforementioned method to make it applicable to all quartic elliptic equations as well. The adaptations needed are described in great detail. Nice examples illustrate the usefulness of this approach. Especially notoriously difficult quartic equations like \(3u^4 - 2v^2=1\), first solved by R. T. Bumby [Math. Scand. 21, 144-148 (1967; Zbl 0169.37402)], and \(u^4-2 u^2+4 = 3v^2\), first solved by W. Ljunggren [Skr. Norske Vid.-Akad., Oslo. I. Math.-Naturvid. Kl. No. 9 (1943; Zbl 0028.00804)], do not cause any computational problem whatsoever, mainly because both represent rank 1 curves and Mordell-Weil bases are easily obtained.

In the present paper the author extends the aforementioned method to make it applicable to all quartic elliptic equations as well. The adaptations needed are described in great detail. Nice examples illustrate the usefulness of this approach. Especially notoriously difficult quartic equations like \(3u^4 - 2v^2=1\), first solved by R. T. Bumby [Math. Scand. 21, 144-148 (1967; Zbl 0169.37402)], and \(u^4-2 u^2+4 = 3v^2\), first solved by W. Ljunggren [Skr. Norske Vid.-Akad., Oslo. I. Math.-Naturvid. Kl. No. 9 (1943; Zbl 0028.00804)], do not cause any computational problem whatsoever, mainly because both represent rank 1 curves and Mordell-Weil bases are easily obtained.

Reviewer: R.J.Stroeker (Rotterdam)