# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 119, 2008, pp. 1–20**

**DOI: 10.4171/RSMUP/119-1**

Integral Points on Certain Elliptic Curves

Hui Lin Zhu^{[1]}and Jian Hua Chen

^{[2]}(1) School of Mathematical Sciences, Xiamen University, 361 005, Fujian, China

(2) School of Mathematics and Statistics, Wuhan University, 430072, Wuhan, Hubei, China

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*x^{2}+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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Zhu Hui Lin, Chen Jian Hua: Integral Points on Certain Elliptic Curves. *Rend. Sem. Mat. Univ. Padova* 119 (2008), 1-20. doi: 10.4171/RSMUP/119-1