Number Theory: A Lively Introduction with Proofs, Applications, and Stories

To the Student.

To the Instructor.

Acknowledgements.

0. Prologue.

1. Numbers, Rational and Irrational.

(Historical figures: Pythagoras and Hypatia).

1.1 Numbers and the Greeks.

1.2 Numbers you know.

1.3 A First Look at Proofs.

1.4 Irrationality of he square root of 2.

1.5 Using Quantifiers.

2. Mathematical Induction.

(Historical figure: Noether).

2.1.The Principle of Mathematical Induction.

2.2 Strong Induction and the Well Ordering Principle.

2.3 The Fibonacci Sequence and the Golden Ratio.

2.4 The Legend of the Golden Ratio.

3. Divisibility and Primes.

(Historical figure: Eratosthenes).

3.1 Basic Properties of Divisibility.

3.2 Prime and Composite Numbers.

3.3 Patterns in the Primes.

3.4 Common Divisors and Common Multiples.

3.5 The Division Theorem.

3.6 Applications of gcd and lcm.

4.The Euclidean Algorithm.

(Historical figure: Euclid).

4.1 The Euclidean Algorithm.

4.2 Finding the Greatest Common Divisor.

4.3 A Greeker Argument that the square root of 2 is Irrational.

5. Linear Diophantine Equations.

(Historical figure: Diophantus).

5.1 The Equation aX + bY = 1.

5.2 Using the Euclidean Algorithm to Find a Solution.

5.3 The Diophantine Equation aX + bY = n.

5.4 Finding All Solutions to a Linear Diophantine Equation.

6. The Fundamental Theorem of Arithmetic.

(Historical figure: Mersenne).

6.1 The Fundamental Theorem.

6.2 Consequences of the Fundamental Theorem.

7. Modular Arithmetic.

(Historical figure: Gauss).

7.1 Congruence modulo n.

7.2 Arithmetic with Congruences.

7.3 Check Digit Schemes.

7.4 The Chinese Remainder Theorem.

7.5 The Gregorian Calendar.

7.6 The Mayan Calendar.

8. Modular Number Systems.

(Historical figure: Turing).

8.1 The Number System Z_n: an Informal View.

8.2 The Number System Z_n: Definition and Basic Properties.

8.3 Multiplicative Inverses in Z_n.

8.4 Elementary Cryptography.

8.5 Encryption Using Modular Multiplication.

9. Exponents Modulo n.

(Historical figure: Fermat).

9.1 Fermat's Little Theorem.

9.2 Reduced Residues and the Euler \phi-function.

9.3 Euler's Theorem.

9.4 Exponentiation Ciphers with a Prime modulus.

9.5 The RSA Encryption Algorithm.

10. Primitive Roots.

(Historical figure: Lagrange).

10.1 Z_n.

10.2 Solving Polynomial Equations in Z_n.

10.3 Primitive Roots.

10.4 Applications of Primitive Roots.

11. Quadratic Residues.

(Historical figure: Eisenstein)

11.1 Squares Modulo n

11.2 Euler's Identity and the Quadratic Character of -1

11.3 The Law of Quadratic Reciprocity

11.4 Gauss's Lemma

11.5 Quadratic Residues and Lattice Points.

11.6 The Proof of Quadratic Reciprocity.

12. Primality Testing.

(Historical figure: Erdös).

12.1 Primality testing.

12.2 Continued Consideration of Charmichael Numbers.

12.3 The Miller-Rabin Primality test.

12.4 Two Special Polynomial Equations in Z_p.

12.5 Proof that Millar-Rabin is Effective.

12.6 Prime Certificates.

12.7 The AKS Deterministic Primality Test.

13. Gaussian Integers.

(Historical figure: Euler).

13.1 Definition of Gaussian Integers

13.2 Divisibility and Primes in Z[i].

13.3 The Division Theorem for the Gaussian Integers.

13.4 Unique Factorization in Z[i].

13.5 Gaussian Primes.

13.6 Fermat's Two Squares Theorem.

14. Continued Fractions.

(Historical figure: Ramanujan).

14.1 Expressing Rational Numbers as Continued Fractions.

14.2 Expressing Irrational Numbers as Continued Fractions.

14.3 Approximating Irrational Numbers Using Continued Fractions.

14.4 Proving that Convergents are Fantastic Approximations.

15. Some Nonlinear Diophantine Equations.

(Historical figure: Germain).

15.1 Pell's Equation

15.2 Fermat's Last Theorem

15.3 Proof of Fermat's Last Theorem for n = 4.

15.4 Germain's Contributions to Fermat's Last Theorem

15.5 A Geometric look at the Equation x⁴ + y⁴ = z².

Appendix: Axioms of Number Theory.

A.1 What is a Number System?

A.2 Order Properties of the Integers.

A.3 Building Results From Our Axioms.

A.4 The Principle of Mathematical Induction.