scholarly journals On the divisor function and class numbers of real quadratic fields, III

1991 ◽  
Vol 67 (10) ◽  
pp. 338-342
Author(s):  
R. A. Mollin ◽  
H. C. Williams
Keyword(s):  
Quadratic Fields ◽  
Class Numbers ◽  
Divisor Function ◽  
Real Quadratic Fields ◽  
Real Quadratic

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.)

EXPONENTS OF CLASS GROUPS OF REAL QUADRATIC FIELDS

2008 ◽  
Vol 04 (04) ◽  
pp. 597-611 ◽  
Author(s):  
KALYAN CHAKRABORTY ◽  
FLORIAN LUCA ◽  
ANIRBAN MUKHOPADHYAY
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Quadratic Number ◽  
Class Groups ◽  
Real Quadratic Fields ◽  
Prime Factors ◽  
Positive Integers ◽  
Real Quadratic

In this paper, we show that the number of real quadratic fields 𝕂 of discriminant Δ𝕂 < x whose class group has an element of order g (with g even) is ≥ x1/g/5 if x > x0, uniformly for positive integers g ≤ ( log log x)/(8 log log log x). We also apply the result to find real quadratic number fields whose class numbers have many prime factors.

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