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Elementary Number Theory

Table of 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

Part Two. Brun's Theorem and Dirichlet's Theorem

• 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

Part Three. Decomposition into Two, Three, and Four Squares

• 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

Part Four. The Class Number of Binary Quadratic Forms

• Introduction
• Factorable and unfactorable forms
• Classes of forms
• The finiteness of the class number
• Primary representations by forms
• The representation of $h(d)$ in terms of $K(d)$
• Gaussian sums; Appendix; Introduction; Kronecker's proof; Schur's proof; Mertens' proof
• Reduction to fundamental discriminants
• The determination of $K(d)$ for fundamental discriminants
• Final formulas for the class number

Appendix. Exercises

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

• Index of conventions; Index of definitions
• Index

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