Integral Points on Certain Elliptic Curves

Abstract

By using algebraic number theory method and p-adic analysis method, we find all integral points on certain elliptic curves

Furthermore, we can find all integer solutions of certain hyperelliptic equations

.
As a particular example, we give a complete solution of the equation which was proposed by Zagier

by this method. In Appendix I and Appendix II, we give the computational method of finding the fundamental unit and factorizing quadratic algebraic number in the subring of a totally complex quartic field, respectively.