目录 Introduction1 A Brief History of Prime 1.1 Euclid and Primes 1.2 Summing Over the Primes 1.3 Listing the Primes 1.4 Fermat Numbers 1.5 Primality Testing 1.6 Proving the Fundamental Theorem of Arithmetic 1.7 Euclid's Theorem Revisited2 Diophantine Equations 2.1 Pythagoras 2.2 The Fundamental Theorem of Arithmetic in Other Contexts 2.3 Sums of Squares 2.4 Siegel's Theorem 2.5 Fermat, Catalan, and Euler3 Quadratic Diophantine Equations 3.1 Quadratic Congruences 3.2 Euler's Criterion 3.3 The Quadratic Reciprocity Law 3.4 Quadratic Rings 3.5 Units in Z 3.6 Quadratic Forms4 Recovering the Fundamental Theorem of Arithmetic 4.1 Crisis 4.2 An Ideal Solution 4.3 Fundamental Theorem of Arithmetic for Ideals 4.4 The Ideal Class Group5 Elliptic Curves 5.1 Rational Points 5.2 The Congruent Number Problem 5.3 Explicit Formulas 5.4 Points of Order Eleven 5.5 Prime Values of Elliptic Divisibility Sequences 5.6 Ramanujan Numbers and the Taxicab Problem6 Elliptic Functions 6.1 Elliptic Functions 6.2 Parametrizing an Elliptic Curve 6.3 Complex Torsion 6.4 Partial Proof of Theorem 6.57 Heights 7.1 Heights on Elliptic Curves 7.2 Mordell's Theorem 7.3 The Weak Mordell Theorem: Congruent Number Curve 7.4 The Parallelogram Law and the Canonical Height 7.5 Mahler Measure and the Naive Parallelogram Law8 The Riemann Zeta Function 8.1 Euler's Summation Formula 8.2 Multiplicative Arithmetic Functions 8.3 Dirichlet Convolution 8.4 Euler Products 8.5 Uniform Convergence 8.6 The Zeta Function Is Analytic 8.7 Analytic Continuation of the Zeta Function9 The Functional Equation of the Riemann Zeta Function 9.1 The Gamma Function 9.2 The Functional Equation 9.3 Fourier Analysis on Schwartz Spaces 9.4 Fourier Analysis of Periodic Functions 9.5 The Theta Function 9.6 The Gamma Function Revisited10 Primes in an Arithmetic Progression 10.1 A New Method of Proof 10.2 Congruences Modulo 3 10.3 Characters of Finite Abelian Groups 10.4 Dirichlet Characters and L-Functions 10.5 Analytic Continuation and Abel's Summation Formula 10.6 Abel's Limit Theorem11 Converging Streams 11.1 The Class Number Formula 11.2 The Dedekind Zeta Function 11.3 Proof of the Class Number Formula 11.4 The Sign of the Gauss Sum 11.5 The Conjectures of Birch and Swinnerton-Dyer12 Computational Number Theory 12.1 Complexity of Arithmetic Computations 12.2 Public-key Cryptography 12.3 Primality Testing: Euclidean Algorithm 12.4 Primality Testing: Pseudoprimes 12.5 Carmichael Numbers 12.6 Probabilistic Primality Testing 12.7 The Agrawal-Kayal-Saxena Algorithm 12.8 Factorizing 12.9 Complexity of Arithmetic in Finite FieldsReferencesIndex Introduction1 A Brief History of Prime 1.1 Euclid and Primes 1.2 Summing Over the Primes 1.3 Listing the Primes 1.4 Fermat Numbers 1.5 Primality Testing 1.6 Proving the Fundamental Theorem of Arithmetic 1.7 Euclid's Theorem Revisited2 Diophantine Equations 2.1 Pythagoras 2.2 The Fundamental Theorem of Arithmetic in Other Contexts 2.3 Sums of Squares 2.4 Siegel's Theorem 2.5 Fermat, Catalan, and Euler3 Quadratic Diophantine Equations 3.1 Quadratic Congruences 3.2 Euler's Criterion 3.3 The Quadratic Reciprocity Law 3.4 Quadratic Rings 3.5 Units in Z 3.6 Quadratic Forms4 Recovering the Fundamental Theorem of Arithmetic 4.1 Crisis 4.2 An Ideal Solution 4.3 Fundamental Theorem of Arithmetic for Ideals 4.4 The Ideal Class Group5 Elliptic Curves 5.1 Rational Points 5.2 The Congruent Number Problem 5.3 Explicit Formulas 5.4 Points of Order Eleven 5.5 Prime Values of Elliptic Divisibility Sequences 5.6 Ramanujan Numbers and the Taxicab Problem6 Elliptic Functions 6.1 Elliptic Functions 6.2 Parametrizing an Elliptic Curve 6.3 Complex Torsion 6.4 Partial Proof of Theorem 6.57 Heights 7.1 Heights on Elliptic Curves 7.2 Mordell's Theorem 7.3 The Weak Mordell Theorem: Congruent Number Curve 7.4 The Parallelogram Law and the Canonical Height 7.5 Mahler Measure and the Naive Parallelogram Law8 The Riemann Zeta Function 8.1 Euler's Summation Formula 8.2 Multiplicative Arithmetic Functions 8.3 Dirichlet Convolution 8.4 Euler Products 8.5 Uniform Convergence 8.6 The Zeta Function Is Analytic 8.7 Analytic Continuation of the Zeta Function9 The Functional Equation of the Riemann Zeta Function 9.1 The Gamma Function 9.2 The Functional Equation 9.3 Fourier Analysis on Schwartz Spaces 9.4 Fourier Analysis of Periodic Functions 9.5 The Theta Function 9.6 The Gamma Function Revisited10 Primes in an Arithmetic Progression 10.1 A New Method of Proof 10.2 Congruences Modulo 3 10.3 Characters of Finite Abelian Groups 10.4 Dirichlet Characters and L-Functions 10.5 Analytic Continuation and Abel's Summation Formula 10.6 Abel's Limit Theorem11 Converging Streams 11.1 The Class Number Formula 11.2 The Dedekind Zeta Function 11.3 Proof of the Class Number Formula 11.4 The Sign of the Gauss Sum 11.5 The Conjectures of Birch and Swinnerton-Dyer12 Computational Number Theory 12.1 Complexity of Arithmetic Computations 12.2 Public-key Cryptography 12.3 Primality Testing: Euclidean Algorithm 12.4 Primality Testing: Pseudoprimes 12.5 Carmichael Numbers 12.6 Probabilistic Primality Testing 12.7 The Agrawal-Kayal-Saxena Algorithm 12.8 Factorizing 12.9 Complexity of Arithmetic in Finite FieldsReferencesIndex
