We discuss a problem a Zagier regarding finding large integral points on elliptic curves, and show how this reduces to a system of polynomial equations. There are four cases where a nontrivial solution is likely to exist, with the easiest being 4 equations in 4 unknowns and the most difficult having 12 of each. The first case was solved in 1988 by Elkies using MACSYMA. We have found a nontrivial solution in the second case using neither Grobner bases nor (multi)resultants, but multidimensional p-adic Newton iteration. We discuss in what circumstances such a technique might be useful.