## 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

### Keywords:

elliptic diophantine equations; unit equation

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### References:

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