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 .
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