Large Integral Points on Elliptic Curves

To ml, friend Dan Shanks Abstract. We describe several methods which permit one to search for big integral points on certain elliptic curves, i.e., for integral solutions (x, ,') of certain Diophantine equations of the form V2 = X3 + ax + b (a, b E Z) in a large range Ixi, Iy I < B, in time polynomial in loglogB. We also give a number of individual examples and of parametric families of examples of specific elliptic curves having a relatively large integral point.