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
y2=(x+a)(x2+bx+c), a,b,cZ, b2<4c.
Furthermore, we can find all integer solutions of certain hyperelliptic equations
Dy2=Ax4+Bx2+C, B2<4AC.
As a particular example, we give a complete solution of the equation which was proposed by Zagier
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