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
_y_2=(x+a)(_x_2+bx+c), a,b,_c_∈Z, b_2<4_c. 
Furthermore, we can find all integer solutions of certain hyperelliptic equations
_D__y_2=_A__x_4+_B_x2+C, B_2<4_AC.
As a particular example, we give a complete solution of the equation which was proposed by Zagier
_y_2=x_3-9_x+28
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.