The catalog description for Theory of Numbers (MATH 5070) in the 2014-15 graduate catalog is: "Divisibility, congruences, quadratic residues, Diophantine equations, and a brief treatment of binary quadratic forms." The course was removed from the catalog in 2015. The 2014-15 ETSU graduate catalog can be found in the ETSU Catalog Archives (accessed 3/7/2022). A better description of this class would be: "Prime numbers, unique factrization, Diophantine equations, elliptic curves and functions, and the Riemann Zeta function." Some exposure to modern algebra, such as ETSU's Introduction to Modern Algebra (MATH 4127/5127), and complex analysis, such as ETSU's Complex Variables (MATH 4337/5337), are prerequisites. Though not a formal prerequisite, background in Elementary Number Theory (MATH 3120) would be useful.

Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer and they contain proofs of the results from the class notes. The "Priintouts of Proofs of Theorems" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).

Introduction.

1. Chapter 1. A Brief History of Prime.
2. Chapter 2. Diophantine Equations.
3. Chapter 3. Quadratic Diophantine Equations.
4. Chapter 4. Recovering the Fundamental Theorem of Arithmetic.
5. Chapter 5. Elliptic Curves.
6. Chapter 6. Elliptic Functions.
7. Chapter 7. Heights.
8. Chapter 8. The Riemann Zeta Function.
9. Chapter 9. The Functional Equation of the Riemann Zeta Function.
10. Chapter 10. Primes in an Arithmetic Progression.
11. Chapter 11. Converging Streams.
12. Chapter 12. Computational Number Theory.

Introduction. Introduction notes

Chapter 1. A Brief History of Prime.

• Section 1.1. Euclid and Primes. Section 1.1 notes
  • Supplement. Proofs of Theorems in Section 1.1. Beamer file of Section 1.1 proofs
  • Supplement. Printout of the Proofs of Theorems in Section 1.1. Print file of Section 1.1 proofs
• Section 1.2. Summing Over the Primes. Section 1.2 notes
  • Supplement. Proofs of Theorems in Section 1.2. Beamer file of Section 1.2 proofs
  • Supplement. Printout of the Proofs of Theorems in Section 1.2. Print file of Section 1.2 proofs
• Section 1.3. Listing the Primes. Section 1.3 notes
• Section 1.4. Fermat Numbers.
• Section 1.5. Primality Testing.
• Section 1.6. Proving the Fundamental Theorem of Arithmetic.
• Section 1.7. Euclid's Theorem Revisited.
• Study Guide 1.

Chapter 2. Diophantine Equations.

• Section 2.1. Pythagoras.
• Section 2.2. The Fundamental Theorem of Arithmetic in Other Contexts.
• Section 2.3. Sums of Squares.
• Section 2.4. Siegel's Theorem.
• Section 2.5. Fermat, Catalan, and Euler.
• Study Guide 2.

Chapter 3. Quadratic Diophantine Equations.

• Section 3.1. Quadratic Congruences.
• Section 3.2. Euler's Criterion.
• Section 3.3. The Quadratic Reciprocity Law.
• Section 3.4. Quadratic Rings.
• Section 3.5. Units in ℤ[√d ], d > 0.
• Section 3.6. Quadratic Forms.
• Study Guide 3.

Chapter 4. Recovering the Fundamental Theorem of Arithmetic.

• Section 4.1. Crisis.
• Section 4.2. An Ideal Solution.
• Section 4.3. Fundamental Theorem of Arithmetic for Ideals.
• Section 4.4. The Ideal Class Group.
• Study Guide 4.

Chapter 5. Elliptic Curves.

• Section 5.1. Rational Points.
• Section 5.2. The Congruent Number Problem.
• Section 5.3. Explicit Formulas.
• Section 5.4. Points of Order Eleven.
• Section 5.5. Prime Values of Elliptic Divisibility Sequences.
• Section 5.6. Ramanujan Numbers and the Taxicab Problem.
• Study Guide 5.

Chapter 6. Elliptic Functions.

• Section 6.1. Elliptic Functions.
• Section 6.2. Parametrizing an Elliptic Curve.
• Section 6.3. Complex Torsion.
• Section 6.4. Partial Proof of Theorem 6.5.
• Study Guide 6.

Chapter 7. Heights.

• Section 7.1. Heights on Elliptic Curves.
• Section 7.2. Mordell's Theorem.
• Section 7.3. The Weak Mordell Theorem: Congruent Number Curve.
• Section 7.4. The Parallelogram Law and the Canonical Height.
• Section 7.5. Mahler Measure and the Naive Prallelogram Law.
• Study Guide 7.

Chapter 8. The Riemann Zeta Function.

• Section 8.1. Euler's Summation Formula.
• Section 8.2. Multiplicative Arithmetic Functions.
• Section 8.3. Dirichlet Convolution.
• Section 8.4. Euler Products.
• Section 8.5. Uniform Convergence.
• Section 8.6. The Zeta Function Is Analytic.
• Section 8.7. Analytic Continuation of the Zeta Function.
• Study Guide 8.

Chapter 9. The Functional Equation of the Riemann Zeta Function.

• Section 9.1. The Gamma Function.
• Section 9.2. The Functional Equation.
• Section 9.3. Fourier Analysis on Schwartz Spaces.
• Section 9.4. Fourier Analysis of Periodic Functions.
• Section 9.5. The Theta Function.
• Section 9.6. The Gamma Function Revisited.
• Study Guide 9.

Chapter 10. Primes in an Arithmetic Progression.

• Section 10.1. A New Method of Proof.
• Section 10.2. Congruences Modulo 3.
• Section 10.3. Characters of Finite Abelian Groups.
• Section 10.4. Dirichlet Characters and L-Functions.
• Section 10.5. Analytic Continuation and Abel's Summation Formula.
• Section 10.6. Abel's Limit Theorem.
• Study Guide 10.

Chapter 11. Coverging Streams.

• Section 11.1. The Class Number Formula.
• Section 11.2. The Dedekind Zeta Function.
• Section 11.3. Proof of the Class Number Formula.
• Section 11.4. The Sign of the Gauss Sum.
• Section 11.5. The Conjecture of Birch and Swinnerton-Dyer.
• Study Guide 11.

Chapter 12. Computational Number Theory.

• Section 12.1. Complexity of Arithmetic Computations.
• Section 12.2. Public-key Cryptography.
• Section 12.3. Primality Testing: Euclidean Algorithm.
• Section 12.4. Primality Testing: Pseudoprimes.
• Section 12.5. Carmichael Numbers.
• Section 12.6. Probabilistic Primality Testing.
• Section 12.7. The Agrawal-Kayal-Sazena Algorithm.
• Section 12.8. Factorizing.
• Section 12.9. Complexity of Arithmetic in Finite Fields.
• Study Guide 12.

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