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Jacobi Sequences of[n2±1](PDF)

《南京师大学报（自然科学版）》[ISSN:1001-4616/CN:32-1239/N]

Issue:
2015年04期
Page:
61-
Research Field:
数学
Publishing date:

Info

Title:
Jacobi Sequences of[n2±1]
Author(s):
He Dawei
School of Mathematical Sciences and Institute of Mathematics，Nanjing Normal University，Nanjing 210023，China
Keywords:
continued fraction; Jacobi symbol; Jacobi sequence
PACS:
O.156.1
DOI:
-
Abstract:
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)].

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.

Memo

Memo:
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Last Update: 2015-12-30