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Solving elliptic diophantine equations avoiding Thue equations and elliptic logarithms. (English) Zbl 0921.11076

The author solves the elliptic equation \[ y^2=(x+p)(x^2+p^2) \tag{1} \] in rational integers \(x,y\) for the primes \(p=167, 223, 337, 1201\). Up to now, elliptic equations have been solved by one of the following two methods: (a) reduce the equation to a finite number of Thue equations and solve the latter using lower bounds for linear forms in (ordinary) logarithms; this involves the computation of the fundamental units of certain number fields; or (b) reduce the equation to an inequality involving elliptic logarithms and solve the latter using lower bounds for linear forms in elliptic logarithms; for this one needs a basis of the Mordell-Weil group of the associated curve.
In the present paper the author applies a third method to (1), suggested to him by Yu. Bilu. Here he reduces (1) to a unit equation with four terms of the shape \[ \gamma \varepsilon^a-\overline{\gamma}\cdot\overline{\varepsilon}^{a} =\overline{\delta}\cdot\overline{\varepsilon}^{-a}-\delta\varepsilon^{-a} \tag{2} \] in the unknown \(a\in{\mathbb Z}\), where \(\varepsilon\) is the fundamental unit of a totally complex quartic field. Supposing \(| \varepsilon| >1\) and \(a\geq 0\), on dividing (2) by \(\overline{\gamma}\overline{\varepsilon}^a\) and taking absolute values one obtains an inequality \(| \beta (\varepsilon /\overline{\varepsilon})^a-1| \ll | \varepsilon | ^{-2a}\) and applying to this a lower bound for linear forms in logarithms one obtains an upper bound for \(a\).

MSC:

11Y50 Computer solution of Diophantine equations
11D25 Cubic and quartic Diophantine equations

Software:

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References:

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