%0 Journal Article
%T 椭圆曲线<i>y</i><sup>2</sup>=(<i>x</i>+2)(<i>x</i><sup>2</sup>-2<i>x</i>+<i>p</i>)的整数点<br>The Integral Points on the Elliptic Curve <i>y</i><sup>2</sup>=(<i>x</i>+2)(<i>x</i><sup>2</sup>-2<i>x</i>+<i>p</i>)
%A 杜先存
%A 赵建红
%A 万飞&lt
%A br&gt
%A DU Xian-cun
%A ZHAO Jian-hong
%A WAN Fei
%J 西南大学学报(自然科学版)
%D 2017
%R 10.13718/j.cnki.xdzk.2017.06.011
%X 利用Legendre符号、同余式、Pell方程的解的性质等初等方法证明了:当<i>p</i>=36<i>s</i><sup>2</sup>-5(<i>s</i>∈<inline-formula>$\mathbb{Z}$</inline-formula><sub>+</sub>, 2？<i>s</i>), 而6<i>s</i><sup>2</sup>-1, 12<i>s</i><sup>2</sup>+1均为素数时, 椭圆曲线<i>y</i><sup>2</sup>=(<i>x</i>+2)(<i>x</i><sup>2</sup>-2<i>x</i>+<i>p</i>)仅有整数点为(<i>x, y</i>)=(-2, 0).<br>Let <i>p</i>=36<i>s</i><sup>2</sup>-5(<i>s</i>∈<inline-formula>$\mathbb{Z}$</inline-formula><sub>+</sub>, 2？<i>s</i>), where is a positive odd number satisfying that 6<i>s</i><sup>2</sup>-1 and 12<i>s</i><sup>2</sup>+1 are primes. It is proved in this paper with the help of the Legendre symbol, congruence and some properties of the solutions to the Pell equation that the elliptic curve <i>y</i><sup>2</sup>=(<i>x</i>+2)(<i>x</i><sup>2</sup>-2<i>x</i>+<i>p</i>) has only integer point (<i>x, y</i>)=(-2, 0)
%K 椭圆曲线
%K 整数点
%K Pell方程
%K Legendre符号
%K 同余&lt
%K br&gt
%K elliptic curve
%K integer point
%K Pell equation
%K Legendre symbol
%K congruence
%U http://xbgjxt.swu.edu.cn/jsuns/html/jsuns/2017/6/201706011.htm