# Integral Points on Certain Elliptic Curves

### Hui Lin Zhu

Xiamen University, Fujian, China### Jian Hua Chen

Wuhan University, Wuhan, Hubei, China

## 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.

## Cite this article

Hui Lin Zhu, Jian Hua Chen, Integral Points on Certain Elliptic Curves. Rend. Sem. Mat. Univ. Padova 119 (2008), pp. 1–20

DOI 10.4171/RSMUP/119-1