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浙江大学学报(理学版)  2016, Vol. 43 Issue (6): 676-678    DOI: 10.3785/j.issn.1008-9497.2016.06.009
数学与计算机科学     
一类超椭圆曲线上的有理点
杨仕椿1,2, 汤建钢2
1. 阿坝师范学院 数学与财经系, 四川 汶川 623000;
2. 伊犁师范学院 数学与统计学院, 新疆 伊宁 835000
Rational points on a class of super elliptic curve
YANG Shichun1,2, TANG Jiangang2
1. Department of Mathematics and Finance, Aba Teachers University, Wenchuan 623000, Sichuan Province, China;
2. College of Mathematics and Statistics, Yili Normal University, Yinning 835000, the Xinjiang Uygur Autonomous Region, China
 全文: PDF(349 KB)  
摘要: p为素数,r≥0是整数.利用广义Fermat方程的深刻结论证明了:若3≤q<100,q≠31,则当p≥5时,超椭圆曲线yp=xx+qr)上仅有平凡的有理点y=0;当q=5,11,23,29,41,47,59,83时,给出了该超椭圆曲线所有的有理点(x,y).特别地,当q=3且r=1时,证明了超椭圆曲线yp=xx+3)仅在p=2时有非平凡的有理点(x,y),并给出了此时所有的非平凡有理点.
关键词: 有理点超椭圆曲线广义Fermat方程    
Abstract: Let p be a prime, and r≥0 be a integer. Using the deeply result of generalized Fermat equation, we prove that if 3≤q<100 and q≠31, then the superelliptic curve yp=x(x+qr) has only ordinary rational point y=0 when p≥5. If q=5,11,23,29,41,47,59,83, we give all of the rational points(x,y) in the superelliptic curve. Furthermore, if q=3 and r=1, the superelliptic curve yp=x(x+3) has a non-trivial rational point(x,y) only when p=2.
Key words: rational point    super elliptic curve    generalized Fermat equation
收稿日期: 2015-09-03 出版日期: 2017-03-07
CLC:  O156.1  
基金资助: 新疆维吾尔自治区普通高等学校重点学科经费资助项目(2012ZDXK21);四川省高等教育人才培养质量教学改革项目(14-156-711);四川省教育厅自然科学研究项目(15ZA0337,15ZB0348,15ZB0350).
通讯作者: 汤建钢,ORCID:http://orcid:org/0000-0001-7662-0394,E-mail:tjg@ylsy.edu.cn     E-mail: tjg@ylsy.edu.cn
作者简介: 杨仕椿(1969-),ORCID:http://orcid.org/0000-0001-5692-7479,男,教授,主要从事数论、组合与编码研究.
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杨仕椿, 汤建钢. 一类超椭圆曲线上的有理点[J]. 浙江大学学报(理学版), 2016, 43(6): 676-678.

YANG Shichun, TANG Jiangang. Rational points on a class of super elliptic curve. Journal of ZheJIang University(Science Edition), 2016, 43(6): 676-678.

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https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2016.06.009        https://www.zjujournals.com/sci/CN/Y2016/V43/I6/676

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