Integral Points on Certain Elliptic Curves
Hui Lin Zhu
Xiamen University, Fujian, ChinaJian 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