Ankeny-Artin-Chowla conjecture and continued fraction expansion

被引：9

作者：

Hashimoto, R ^{[1
]}

机构：

[1] Nagoya Univ, Grad Sch Human Informat, Nagoya, Aichi 4648601, Japan

来源：

JOURNAL OF NUMBER THEORY | 2001年 / 90卷 / 01期

关键词：

quadratic field; fundamental unit; continued fraction;

D O I：

10.1006/jnth.2001.2652

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

For any prime p congruent to I modulo 4, let (t + u rootp)/2 be the fundamental unit of Q(rootp). Then Ankeny, Artin, and Chowla conjectured that it is not divisible by p. In this paper, we investigate a certain relation between the conjecture and the continued fraction expansion of (1 + rootp)/2, Consequently, we prove that the conjecture is true if p is not "small" in some sense. (C) 2001 Academic Press.

引用

页码：143 / 153

页数：11

共 8 条

[1] THE CLASS-NUMBER OF REAL QUADRATIC NUMBER FIELDS [J].

ANKENY, NC ;

ARTIN, E ;

CHOWLA, S .

ANNALS OF MATHEMATICS, 1952, 56 (03) :479-493

[2] ON CONTINUED FRACTIONS OF GIVEN PERIOD [J].

FRIESEN, C .

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 1988, 103 (01) :9-14

[3]

MORDELL LJ, 1960, ACTA ARITH, V6, P137

[4]

PERRON O, 1954, LEHRE KETTENBUCH, V1

[5]

Ribenboim P., 1996, The New Book of Prime Number Records

[6] Explicit representation of fundamental units of some real quadratic fields .2. [J].

Tomita, K .

JOURNAL OF NUMBER THEORY, 1997, 63 (02) :275-285

[7]

VANDPOORTEN AJ, 2000, IN PRESS MATH COMP

[8] THE FUNDAMENTAL UNIT AND BOUNDS FOR CLASS-NUMBERS OF REAL QUADRATIC FIELDS [J].

YOKOI, H .

NAGOYA MATHEMATICAL JOURNAL, 1991, 124 :181-197

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