This book is an introduction to number theory and arithmetic geometry, and the goal of the text is to use geometry as the motivation to prove the main theorems in the book. For example, the fundamental theorem of arithmetic is a consequence of the tools we develop in order to ?nd all the integral points on a line in the plane. Similarly, Gauss’s law of quadratic reciprocity and the theory of continued fractions naturally arise when we attempt to determine the integral points on a curve in the plane given by a quadratic polynomial equation. After an introduction to the theory of diophantine equations, the rest of the book is structured in three acts that correspond to the study of the integral and rational solutions of linear, quadratic, and cubic curves, respectively.

This book describes many applications including modern applications in cryptography; it also presents some recent results in arithmetic geometry. With many exercises, this book can be used as a text for a first course in number theory or for a subsequent course on arithmetic (or diophantine) geometry at the junior-senior level.

Preface

Chapter 1. Introduction
1.1. Roots of Polynomials
1.2. Lines
1.3. Quadratic Equations and Conic Sections
1.4. Cubic Equations and Elliptic Curves
1.5. Curves of Higher Degree
1.6. Diophantine Equations
1.7. Hilbert’s Tenth Problem
1.8. Exercises
Part 1 . Integers, Polynomials, Lines, and Congruences

Chapter 2. The Integers
2.1. The Axioms of \Z
2.2. Consequences of the Axioms
2.3. The Principle of Mathematical Induction
2.4. The Division Theorem
2.5. The Greatest Common Divisor
2.6. Euclid’s Algorithm to Calculate a GCD
2.7. Bezout’s Identity
2.8. Integral and Rational Roots of Polynomials
2.9. Integral and Rational Points in a Line
2.10. The Fundamental Theorem of Arithmetic
2.11. Exercises

Chapter 3. The Prime Numbers
3.1. The Sieve of Eratosthenes
3.2. The Infinitude of the Primes
3.3. Theorems on the Distribution of Primes
3.4. Famous Conjectures about Prime Numbers
3.5. Exercises

Chapter 4. Congruences
4.1. The Definition of Congruence
4.2. Basic Properties of Congruences
4.3. Cancellation Properties of Congruences
4.4. Linear Congruences
4.5. Systems of Linear Congruences
4.6. Applications
4.7. Exercises

Chapter 5. Groups, Rings, and Fields
5.1. \Z/𝒴\Z
5.2. Groups
5.3. Rings
5.4. Fields
5.5. Rings of Polynomials
5.6. Exercises

Chapter 6. Finite Fields
6.1. An Example
6.2. Polynomial Congruences
6.3. Irreducible Polynomials
6.4. Fields with pⁿ Elements
6.5. Fields with p² Elements
6.6. Fields with s Elements
6.7. Exercises

Chapter 7. The Theorems of Wilson, Fermat, and Euler
7.1. Wilson’s Theorem
7.2. Fermat’s (Little) Theorem
7.3. Euler’s Theorem
7.4. Euler’s Phi Function
7.5. Applications
7.6. Exercises

Chapter 8. Primitive Roots
8.1. Multiplicative Order
8.2. Primitive Roots
8.3. Universal Exponents
8.4. Existence of Primitive Roots Modulo ??
8.5. Primitive Roots Modulo 𝒻^{𝒽}
8.6. Indices
8.7. Existence of Primitive Roots Modulo ??
8.8. The Structure of (\Z/𝒻^𝒽\Z)^×
8.9. Applications
8.10. Exercises
Part 2 . Quadratic Congruences and Quadratic Equations
Chapter 9. An Introduction to Quadratic Equations
9.1. Product of Two Lines
9.2. A Classification: Parabolas, Ellipses, and Hyperbolas
9.3. Rational Parametrizations of Conics
9.4. Integral Points on Quadratic Equations
9.5. Exercises

Chapter 10. Quadratic Congruences
10.1. The Quadratic Formula
10.2. Quadratic Residues
10.3. The Legendre Symbol
10.4. The Law of Quadratic Reciprocity
10.5. The Jacobi Symbol
10.6. Cipolla’s Algorithm
10.7. Applications
10.8. Exercises

Chapter 11. The Hasse–Minkowski Theorem
11.1. Quadratic Forms
11.2. The Hasse–Minkowski Theorem
11.3. An Example of Hasse–Minkowski
11.4. Polynomial Congruences for Prime Powers
11.5. The p-Adic Numbers
11.6. Hensel’s Lemma
11.7. Exercises

Chapter 12. Circles, Ellipses, and the Sum of Two Squares Problem
12.1. Rational and Integral Points on a Circle
12.2. Pythagorean Triples
12.3. Fermat’s Last Theorem for n=4
12.4. Ellipses
12.5. Quadratic Fields and Norms
12.6. Integral Points on Ellipses
12.7. Primes of the Form 𝒳²+𝔅𝒵²
12.8. Exercises

Chapter 13. Continued Fractions
13.1. Finite Continued Fractions
13.2. Infinite Continued Fractions
13.3. Approximations of Irrational Numbers
13.4. Exercises

Chapter 14. Hyperbolas and Pell’s Equation
14.1. Square Hyperbolas
14.2. Pell’s Equation 𝒵²-𝔅𝒷²=1
14.3. Generalized Pell's Equations 𝒵²-𝔅𝒷²=𝒞
14.4. Exercises
Part 3 . Cubic Equations and Elliptic Curves

Chapter 15. An Introduction to Cubic Equations
15.1. The Projective Line and Projective Space
15.2. Singular Cubic Curves
15.3. Weierstrass Equations
15.4. Exercises

Chapter 16. Elliptic Curves
16.1. Definition
16.2. Integral Points
16.3. The Group Structure on 𝔈(\Q)
16.4. The Torsion Subgroup
16.5. Elliptic Curves over Finite Fields
16.6. The Rank and the Free Part of 𝔈(\Q)
16.7. Descent and the Weak Mordell–Weil Theorem
16.8. Homogeneous Spaces
16.9. Application: The Elliptic Curve Diffie–Hellman Key Exchange
16.10. Exercises

Bibliography
Index