[1]朱德辉.关于不定方程 x2-3y4=166[J].重庆师范大学学报(自然科学版),2008,25(03):21.[doi:10.11721/cqnuj20080306]
ZHU De-hui.On the Diophantine Equation x2-3y4=166[J].期刊社,2008,25(03):21.[doi:10.11721/cqnuj20080306]
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重庆师范大学学报(自然科学版)[ISSN:1672-6693/CN:50-1165/N]
- 卷:
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25
- 期数:
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2008年03期
- 页码:
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21
- 栏目:
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理论与应用研究
- 出版日期:
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2008-07-25
文章信息/Info
- Title:
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On the Diophantine Equation x2-3y4=166
- 作者:
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朱德辉
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重庆师范大学 数学与计算机科学学院,重庆400047
- Author(s):
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ZHU De-hui
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College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China
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- 关键词:
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平方剩余; 递归序列; 正整数解; 不定方程
- Keywords:
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congruence; recursive sequence; quadratic remainder; positive integral solution
- 分类号:
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- DOI:
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10.11721/cqnuj20080306
- 文献标志码:
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A
- 摘要:
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利用一种初等的证明方法,对一个不定方程 x 2 -3y4=166 的正整数解进行了研究。证明过程中仅涉及到初等的数论知识,即运用递归数列、同余式和平方剩余的方法。首先利用 Pell 方程的解的性质把不定方程 x 2 -3y4=166 的解转化为由两个非结合类给出,然后再进一步利用相关知识使得问题简化为两种相对简单的情况,对其每一种情况都利用递归数列,同余式和平方剩余的相关知识对其是否有正整数解进行证明,如果有正整数解则进行求解。最后得出该不定方程 x 2 -3y4=166 仅有正整数解( x , y ) = ( 13 , 1 ) , ( 293 , 13 )。
- Abstract:
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The study of the Diophantine equation x2-Dy4=N ( D and N are the given integers, D > 0 and D is non-square ) has caused some authors’ interests, such as Cohn, Tzanakis, LI Jin-xiang , LIN LI-juan. Cohn has proven some conclusions. For example: N(5, 44)=1 , (x, y)=(7 , 1); N(5 , 11)=2, (x, y)=(4,1) , (56,5); N(5 , -44)=3 , (x, y)=(6 , 2) , (19 , 3) , (181 , 9). Tzanakis has proven some conclusions while y≡0 (mod8). For example: N(2, 17)=0, N(2, 41)=0, N( 8 , 17)=0, N(2, 97)=0. LI Jin-xiang hasproven one conclusion: N(3, 46)=2, (x, y)=(7,1) , (17, 3). LIN Li-juan has also proven one conclusion: N(3, 22)=2 , (x, y)=(5, 1) , (85, 7). But this Diophantine equation x2-3y4=166 still has not been solved until now. In this paper the author has proved that the Diophantine equation x2-3y4=166 has only positive integral solutions(x, y)=(13 , 1) , (293 , 13) with the primary methods of recursive sequence , quadratic remainder and congruence.
备注/Memo
- 备注/Memo:
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收稿日期: 2007-12-10 修回日期:2008-02-20
资助项目:重庆市教委科研基金项目(No. 010204)
作者简介:朱德辉(1981-),女,硕士研究生,研究方向为数论
更新日期/Last Update:
2008-07-22