JournalsrsmupVol. 119pp. 1–20

Integral Points on Certain Elliptic Curves

  • Hui Lin Zhu

    Xiamen University, Fujian, China
  • Jian Hua Chen

    Wuhan University, Wuhan, Hubei, China
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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