Quadratics

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