I could not manage to find a reasonable link to Runge's method for diophantine equations (the wiki page on Runge contains only a mention); my memory says that it is in Mordell's book (at least in relation with Catalan's equation). The idea of the method for solving the diophantine equation $y^m=F(x)$, where $F(x)$ is a polynomial, is to use the binomial theorem for $\sqrt[m]{F(x)}$ and truncate the tail which is less than $1/2$.