S-integral Solutions to a Weierstrass Equation

The rational solutions with as denominators powers of to the elliptic diophantine equation y x x are determined An idea of Yuri Bilu is applied which avoids Thue and Thue Mahler equations and deduces four term S unit equations with special properties that are solved by linear forms in real and p adic logarithms Introduction In a recent paper SW my colleague R J Stroeker and I determined the complete set of solutions in rational integers to the diophantine equation y x x Q The quartic diophantine equation Q can be seen as a model of an elliptic curve In our solution methods we did not use that viewpoint but rather worked algebraically Recently N Tzanakis T has shown how the elliptic logarithms method of Stroeker and Tzanakis ST originally designed for Weierstrass models of elliptic curves only can be adapted to the situation of a quartic model for an elliptic curve and in fact he chose equation Q as one of the examples to illustrate his ideas We stress that the problem of integral points on elliptic curves is not a well de ned problem in contrast to the problem of rational points Bi rational transformations between di erent models of the same curve do respect the concept of rational point but not that of integral point in an This research was supported by the Netherlands Mathematical Research Foundation SWON with nancial aid from the Netherlands Organization for Scienti c Research NWO Manuscrit re cu le mars Benjamin M M de Weger essential way Therefore one should speak of integral solutions to elliptic equations rather than of integral points on elliptic curves When Stroeker and I worked on equation Q we also computed as kind of a standard procedure when dealing with an elliptic curve the Weierstrass equation of the elliptic curve it represents being