Lengths of the periods of the continued fraction expansion of quadratic irrationalities and on the class numbers of real quadratic fields

1991 ◽  
Vol 53 (3) ◽  
pp. 229-237
Author(s):  
E. P. Golubeva
Keyword(s):  
Continued Fraction ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Continued Fraction Expansion ◽  
Real Quadratic Fields ◽  
Real Quadratic

Class Numbers of Real Quadratic Fields, Continued Fractions, Reduced Ideals, Prime-Producing Quadratic Polynomials and Quadratic Residue Covers

1992 ◽  
Vol 44 (4) ◽  
pp. 824-842 ◽  
Author(s):  
S. Louboutin ◽  
R. A. Mollin ◽  
H. C. Williams
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Quadratic Residue ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Continued Fraction Expansion ◽  
Quadratic Polynomials ◽  
Real Quadratic Fields ◽  
The Relationship ◽  
Real Quadratic

AbstractIn this paper we consider the relationship between real quadratic fields, their class numbers and the continued fraction expansion of related ideals, as well as the prime-producing capacity of certain canonical quadratic polynomials. This continues and extends work in [10]–[31] and is related to work in [3]–[4].

scholarly journals Lower bounds for fundamental units of real quadratic fields

2002 ◽  
Vol 166 ◽  
pp. 29-37 ◽  
Author(s):  
Koshi Tomita ◽  
Kouji Yamamuro
Keyword(s):  
Lower Bounds ◽  
Continued Fraction ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Real Quadratic Field ◽  
Integral Basis ◽  
Continued Fraction Expansion ◽  
Real Quadratic Fields ◽  
Fundamental Units ◽  
Real Quadratic

AbstractLet d be a square-free positive integer and l(d) be the period length of the simple continued fraction expansion of ωd, where ωd is integral basis of ℤ[]. Let εd = (td + ud)/2 (> 1) be the fundamental unit of the real quadratic field ℚ(). In this paper new lower bounds for εd, td, and ud are described in terms of l(d). The lower bounds of εd are sharper than the known bounds and those of td and ud have been yet unknown. In order to show the strength of the method of the proof, some interesting examples of d are given for which εd and Yokoi’s d-invariants are determined explicitly in relation to continued fractions of the form .

scholarly journals The fundamental unit and bounds for class numbers of real quadratic fields

1991 ◽  
Vol 124 ◽  
pp. 181-197 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Class Number ◽  
Continued Fraction ◽  
Diophantine Equations ◽  
Fundamental Unit ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Imaginary Quadratic Fields ◽  
Real Quadratic Fields ◽  
Real Quadratic

Although class number one problem for imaginary quadratic fields was solved in 1966 by A. Baker [3] and by H. M. Stark [25] independently, the problem for real quadratic fields remains still unsettled. However, since papers by Ankeny–Chowla–Hasse [2] and H. Hasse [9], many papers concerning this problem or giving estimate for class numbers of real quadratic fields from below have appeared. There are three methods used there, namely the first is related with quadratic diophantine equations ([2], [9], [27, 28, 29, 31], [17]), and the second is related with continued fraction expantions ([8], [4], [16], [14], [18]).

scholarly journals Distribution of Class Numbers in Continued Fraction Families of Real Quadratic Fields

2018 ◽  
Vol 61 (4) ◽  
pp. 1193-1212
Author(s):  
Alexander Dahl ◽  
Vítězslav Kala
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Random Model ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Real Quadratic Fields ◽  
Continued Fraction Expansions ◽  
Real Quadratic

AbstractWe construct a random model to study the distribution of class numbers in special families of real quadratic fields ${\open Q}(\sqrt d )$ arising from continued fractions. These families are obtained by considering continued fraction expansions of the form $\sqrt {D(n)} = [f(n),\overline {u_1,u_2, \ldots ,u_{s-1} ,2f(n)]} $ with fixed coefficients u1, …, us−1 and generalize well-known families such as Chowla's 4n2 + 1, for which analogous results were recently proved by Dahl and Lamzouri [‘The distribution of class numbers in a special family of real quadratic fields’, Trans. Amer. Math. Soc. (2018), 6331–6356].

scholarly journals Some Criteria for Class Numbers to Be Non-One

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Ahmad Issa ◽  
Hasan Sankari
Keyword(s):  
Positive Integer ◽  
Continued Fraction ◽  
Period Length ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Pell Equation ◽  
Fixed Integer ◽  
Real Quadratic Fields ◽  
Continued Fraction Expansions ◽  
Real Quadratic

Let d be a positive integer which is not a perfect square and n be any nonzero fixed integer. Then, the equation x 2 − d y 2 = n is known as the general Pell equation. In this paper, we give some criteria for class numbers of certain real quadratic fields to be greater than one, depending on the solvability of the general Pell equation, ideals in quadratic orders, and the period length of the simple continued fraction expansions of d .

scholarly journals On the Insolubility of a Class of Diophantine Equations and the Nontriviality of the Class Numbers of Related Real Quadratic Fields of Richaud-Degert Type

1987 ◽  
Vol 105 ◽  
pp. 39-47 ◽  
Author(s):  
R. A. Mollin
Keyword(s):  
Diophantine Equations ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Real Quadratic Fields ◽  
Integer Solutions ◽  
The Relationship ◽  
Real Quadratic

Many authors have studied the relationship between nontrivial class numbers h(n) of real quadratic fields and the lack of integer solutions for certain diophantine equations. Most such results have pertained to positive square-free integers of the form n = l2 + r with integer >0, integer r dividing 4l and — l<r<l. For n of this form, is said to be of Richaud-Degert (R-D) type (see [3] and [8]; as well as [2], [6], [7], [12] and [13] for extensions and generalizations of R-D types.)

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