scholarly journals On the Fundamental Units and the Class Numbers of Real Quadratic Fields II

1987 ◽  
Vol 10 (2) ◽  
pp. 259-270 ◽  
Author(s):  
Takashi AZUHATA
Keyword(s):  
Quadratic Fields ◽  
Class Numbers ◽  
Real Quadratic Fields ◽  
Fundamental Units ◽  
Real Quadratic

scholarly journals On the fundamental units and the class numbers of real quadratic fields

1984 ◽  
Vol 95 ◽  
pp. 125-135 ◽  
Author(s):  
Takashi Azuhata
Keyword(s):  
Rational Number ◽  
Quadratic Field ◽  
Sufficient Conditions ◽  
Prime Numbers ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Real Quadratic Field ◽  
Real Quadratic Fields ◽  
Fundamental Units ◽  
Real Quadratic

Let Q be the rational number field and h(m) be the class number of the real quadratic field with a positive square-free integer m. It is known that if h(m) = 1 holds, then m is one of the following four types with prime numbers p ≡ 1, pt ≡ 3 (mod 4) (1 昤 i ≥ 4) : i) m = p, ii) m = p1, iii) m = 2 or m = 2p2, iv) m = p3p4 (see Behrbohm and Rédei [1]). The sufficient conditions for h(m) > 1 with these m were given by several authors: in all cases by Hasse [2], in case i) by Ankeny, Chowla and Hasse [3] and by Lang [4], in case ii) by Takeuchi [5] and by Yokoi [6].

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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