Integral Points on Certain Elliptic Curves

Hui Lin Zhu; Jian Hua Chen

doi:10.4171/rsmup/119-1

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} + b x + c), a, b, c \in Z, b^{2} < 4 c .

Furthermore, we can find all integer solutions of certain hyperelliptic equations

D y^{2} = A x^{4} + B x^{2} + C, B^{2} < 4 A C

.
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