椭圆曲线y2=x3+27x-62的整数点

吴华明

数学学报 ›› 2010, Vol. 53 ›› Issue (1) : 205-208.

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数学学报 ›› 2010, Vol. 53 ›› Issue (1) : 205-208. DOI: 10.12386/A2010sxxb0025
论文

椭圆曲线y2=x3+27x-62的整数点

    吴华明
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Points on the Elliptic Curve y2=x3+27x-62

    Hua Ming WU
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摘要

根据四次Diophantine方程的已知结果,运用初等数论方法证明了:椭圆曲线y2=x3+27x-62仅有整数点(x, y)=(2, 0)和(28844402,±154914585540).

 

Abstract

Using some known results of quartic diophantine equations with elementary number theory methods, we prove that the elliptic curve y2=x3+27x-62 has only the integral points (x, y)=(2, 0) and (28844402, ±154914585540).

 

关键词

椭圆曲线 / 整数点 / Pell方程

Key words

elliptic curve / integral point / Pell equation

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吴华明. 椭圆曲线y2=x3+27x-62的整数点. 数学学报, 2010, 53(1): 205-208 https://doi.org/10.12386/A2010sxxb0025
Hua Ming WU. Points on the Elliptic Curve y2=x3+27x-62. Acta Mathematica Sinica, Chinese Series, 2010, 53(1): 205-208 https://doi.org/10.12386/A2010sxxb0025

参考文献


[1] Silverman J. H., The Arithmetic of Elliptic Curves, New York: Springer Verlag, 1999.

[2] Zhu H. L., Chen J. H., A note on two diophantine equation y2=nx(x2±1), Acta Mathematica Sinica, Chinese Series, 2007, 50(5): 1071--1074.

[3] Zagier D., Large integral point on elliptic curves, Math Comp., 1987, 48(177): 425--536.

[4] Zhu H. L., Chen J. H., lntegral points on y2=x3+27x-62, J. Math. Study, 2009, 42(2): 117--125.

[5] Min S. H., Yan S. J., Elementary Number Theory, Beijing: Higher Education Press, 2003, 163--166 (in Chinese).

[6] Walsh G., A note on a theorem of Ljunggren and the diophantine equations x2-kxy2+y4=1 or 4, Arch. Math., 1999, 73(2): 119--125.

[7] Walker D. T., On the diophantine equation mX2-nY2=±1, Amer, Math. Monthly, 1967, 74(6): 504--513.

基金

国家自然科学基金资助项目(10771186);广东省自然科学基金项目(06029035)

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