@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}, }