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List of Symbols
Preface
Introduction
Background from Algebraic Number Theory
Quadratic Fields: Integers and Units
The Arithmetic of Ideals in Quadratic Fields
The Class Group and Class Number
Reduced Ideals
Quadratic Orders
Powerful Numbers: An Application of Real Quadratics
Continued Fractions Applied to Quadratic Fields
Continued Fractions and Real Quadratics: The Infrastructure
The Continued Fraction Analogue for Complex Quadratics
Diophantine Equations and Class Numbers
Class Numbers and Complex Quadratics
Real Quadratics and Diophantine Equations
Reduced Ideals and Diophantine Equations
Class Numbers and Real Quadratics
Halfway to a Solution
Prime-Producing Polynomials
Complex Prime-Producers
Real Prime-Producers
Density of Primes
Class Numbers: Criteria and Bounds
Factoring Rabinowitsch
Class Number One Criteria
Class Number Bounds via the Divisor Function
The GRH: Relevance of the Riemann Hypothesis
Ambiguous Ideals
Ambiguous Cycles in Real Orders: The Palindromic Index
Exponent Two
Influence of the Infrastructure
Quadratic Residue Covers
Consecutive Powers
Algorithms
Computation of the Class Number of a Real Quadratic Field
Cryptology
Implications of Computational Mathematics for the Philosophy of Mathematics
Appendix A: Tables
Table A1: This is a list of all positive fundamental radicands with class number h? = 1 and period length l , of the simple continued fraction expansion of the principal class, less then 24. Table A8 is known to be unconditionally complete whereas Table A1 is complete with one GRH-ruled out exception, as are Tables A2-A4, A6-A7 and A9
Table A2: This is a subset of Table A1 with D ? 1 (mod 8)
Table A3: h? = 2 for fundamental radicands D > 0 with l ? 24
Table A4: This is a list of all fundamental radicands of ERD-type with class groups of exponent 2, broken down into three parts depending on congruence modulo 4 of the radicand
Table A5: This three-part table is an illustration of a computer run done for the proof of Theorem 6.2.2
Table A6: This is a list of all fundamental radicands D > 0 of ERD-type having no split primes less than the Minkowski bound
Table A7: This is a complete list of all fundamental radicands D > 0 with n? ? 0 (see Exercise 3.2.11) and associated regulators, such that the class number is 1
Table A8: This is a list of all fundamental discriminants D ? 1 (mod 8) of ERD-type with class number less than 24, and is known to be unconditionally complete
Table A9: This table lists all fundamental discriminants of ERD-type with class number 2
Appendix B: Fundamental Units of Real Quadratic Fields
	This list is broken up into three parts according to congruence modulo 4 of fundamental radicands less than 2 . 103
Appendix C: Class Numbers of Real Quadratic Fields
	This table is presented in matrix form with each entry describing a specified class number together with the norm of the fundamental unit with radicands less than 104
Appendix D: Class Numbers of Complex Quadratic Fields (and their class group structure)
	This is a table of fundamental radicands D

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