Class number formula for certain imaginary quadratic fields

L - Functions and Class Numbers of Imaginary Quadratic Fields and of Quadratic Extensions of an Imaginary Quadratic Field

Starting from the analytic class number formula involving its Lfunction, we first give an expression for the class number of an imaginary quadratic field which, in the case of large discriminants, provides us with a much more powerful numerical technique than that of counting the number of reduced definite positive binary quadratic forms, as has been used by Buell in order to compute his class ...

Class Numbers of Quadratic Fields Determined by Solvability of Diophantine Equations

In the literature there has been considerable attention given to the exploration of relationships between certain diophantine equations and class numbers of quadratic fields. In this paper we provide criteria for the insolvability of certain diophantine equations. This result is then used to determine when related real quadratic fields have class number bigger than 1. Moreover, based on criteri...

Lectures on the Dirichlet Class Number Formula for Imaginary Quadratic Fields

Indivisibility of class numbers of imaginary quadratic fields

We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to −X whose class numbers are indivisible by a given prime and whose imaginary quadratic fields satisfy any given set of local conditions. This estimate matches the best results in the direction of the Cohen–Lenstra heuristics for ...

On the real quadratic fields with certain continued fraction expansions and fundamental units

The purpose of this paper is to investigate the real quadratic number fields $Q(sqrt{d})$ which contain the specific form of the continued fractions expansions of integral basis element  where $dequiv 2,3( mod  4)$ is a square free positive integer. Besides, the present paper deals with determining the fundamental unit$$epsilon _{d}=left(t_d+u_dsqrt{d}right) 2left.right > 1$$and  $n_d$ and $m_d...

Diophantine Equations and Class Numbers

The goals of this paper are to provide: (I ) sufficient conditions, based on the solvability of certain diophantine equations, for the non-triviality of the dass numbers of certain real quadratic fields; (2) sufficient conditions for the divisibility of the class numbers of certain imaginary quadratic fields by a given integer; and (3) necessary and sufficient conditions for an algebraic intege...