@article {IOPORT.01247717,
    author       = {de Weger, Benjamin M.M.},
    title        = {Solving elliptic diophantine equations avoiding Thue equations and elliptic logarithms.},
    year         = {1998},
    journal      = {Experimental Mathematics},
    volume       = {7},
    number       = {3},
    issn         = {1058-6458},
    pages        = {243-256},
    publisher    = {Taylor \& Francis, Philadelphia, PA},
    doi          = {10.1080/10586458.1998.10504371},
    abstract     = {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 $a\in{\Bbb Z}$, 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 $a$.},
    reviewer     = {Jan-Hendrik Evertse (Leiden)},
    identifier   = {01247717},
}