[1]何达伟.[n2±1]的雅可比序列(英文)[J].南京师范大学学报(自然科学版),2015,38(04):61.
He Dawei.Jacobi Sequences of[n2±1][J].Journal of Nanjing Normal University(Natural Science Edition),2015,38(04):61.
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[n2±1]的雅可比序列(英文)(
)
《南京师范大学学报》(自然科学版)[ISSN:1001-4616/CN:32-1239/N]
- 卷:
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第38卷
- 期数:
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2015年04期
- 页码:
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61
- 栏目:
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数学
- 出版日期:
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2015-12-30
文章信息/Info
- Title:
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Jacobi Sequences of[n2±1]
- 作者:
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何达伟
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南京师范大学数学科学学院,江苏 南京 210023
- Author(s):
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He Dawei
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School of Mathematical Sciences and Institute of Mathematics,Nanjing Normal University,Nanjing 210023,China
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- 关键词:
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连分数; 雅可比符号; 雅可比序列
- Keywords:
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continued fraction; Jacobi symbol; Jacobi sequence
- 分类号:
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O.156.1
- 文献标志码:
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A
- 摘要:
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令[pk/qk][(k≥0)]表示无理数[θ]的连分数展开式的第[k]个渐近分数. 我们研究雅可比序列[(pk/qk)][(k≥0)]. K. Girstmair证明了当[θ=e]时,此序列是周期长度为24的纯周期序列;当[θ=e2]时,此序列是周期长度为40的纯周期序列. 类似地,本文我们分别确定了[θ=n2+1][(n≥1)] 和[θ=n2-1][(n≥2)] 的雅可比序列的周期长度.
- Abstract:
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Let[pk/qk][(k≥0)]be the[k]th convergent of the continued fraction expansion of an irrational real number[θ]. We investigate the sequence of Jacobi symbols[(pk/qk)][(k≥0)]. K. Girstmair showed that this sequence is purely periodic with period length 24 for [θ=e] and period length 40 for [θ=e2.] Similarly,in this paper,we determine the period lengths of the Jacobi sequences for [θ=n2+1][(n≥1)] and [θ=n2-1][(n≥2)].
参考文献/References:
[1]GIRSTMAIR K. Continued fractions and Jacobi symbols[J]. Int J Number Theory,2011,7:1 543-1 555.
[2]GIRSTMAIR K. Periodic continued fractions and Jacobi symbols[J]. Int J Number Theory,2012,8:1 519-1 525.
[3]GIRSTMAIR K. Jacobi symbols and Euler’s number e[J]. J Number Theory,2014,135:155-166.
[4]SIERPINSKI W. Elementary theory of numbers[M]. Warszawa:North-Holland PWN-Polish Scientific Publishers,1988.
[5]HUA L K. Introduction to number theory[M]. Berlin:Springer,1982.
备注/Memo
- 备注/Memo:
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收稿日期:2015-04-20.
基金项目:国家自然科学基金(11371195).
通讯联系人:何达伟,硕士研究生,研究方向:数论,E-mail:m15212242516_1@163.com
更新日期/Last Update:
2015-12-30