Elementary number theory

Contents

Part One. Foundations of Number Theory

• The greatest common divisor of two numbers
• Prime numbers and factorization into prime factors
• The greatest common divisor of several numbers
• Number-theoretic functions
• Congruences
• Quadratic residues
• Pell's equation

• Introduction
• Some elementary inequalities of prime number theory
• Brun's theorem on prime pairs
• Dirichlet's theorem on the prime numbers in an arithmetic progression; Further theorems on congruences; Characters; $L$-series; Dirichlet's proof

• Introduction
• Farey fractions
• Decomposition into two squares
• Decomposition into four squares; Introduction; Lagrange's theorem; Determination of the number of solutions
• Decomposition into three squares; Equivalence of quadratic forms; A necessary condition for decomposability into three squares; The necessary condition is sufficient

1. Introduction
2. Factorable and unfactorable forms
3. Classes of forms
4. The finiteness of the class number
5. Primary representations by forms
6. The representation of $h(d)$ in terms of $K(d)$
7. Gaussian sums; Appendix; Introduction; Kronecker's proof; Schur's proof; Mertens' proof
8. Reduction to fundamental discriminants
9. The determination of $K(d)$ for fundamental discriminants
10. Final formulas for the class number

• Exercises for part one
• Exercises for part two
• Exercises for part three

Index