The following is a tentative syllabus for the course.
| Week | Lectures | Sections in Text | Brief Description | Suggested Practice |
|---|---|---|---|---|
| 1 | 9/16 | 1.1 | Introduction, Division and Euclidean Algorithms | Prove divisibility facts 5, 7, and 9. Read over the proof of Thm 1.1 in the text, and note any questions you have. |
| 9/18 | 1.2, 1.3 | Bezout's identity, least common multiples | Write gcd(1485, 1745) in the form 1485u+1745v. Find lcm(1485, 1745). Show that c|a and c|b iff c|gcd(a,b). |
|
| 9/20 | 1.3, 1.4 | LCMs, Linear Diophantine Equations | Find the general solution to 1066x+1492y=8. Exercise 1.25 on p. 17 of the text. |
|
| 2 | 9/23 | 2.1, 2.2 | Fundamental Theorem of Arithmetic, Distribution of primes | Prove the following: if a positive integer n is not a perfect square, then the square root of n is an irrational number. |
| 9/24 | (x-hour) Appendix A | Induction practice | Solutions to practice problems | |
| 9/25 | 2.2, 2.4 | Distribution of primes, primality testing | Exercise 2.7 on p.29 of the text. Use the Sieve of Eratosthenes to find all the primes less than 100. |
|
| 9/27 | 3.1 | Modular arithmetic | Find the least absolute and least non-negative residues of 19*14 mod (23). // Prove the divisibility rule for 5. | |
| 3 | 9/30 | 3.2 | Linear Congruences (Clarification) | Exercise 3.7 on p.52 of the text. |
| 10/1 | (x-hour) 10 ways | Proof techniques | Solutions to practice problems | |
| 10/2 | 3.3, 3.4 | Chinese Remainder Theorem | Exercises 3.9, 3.11, and 3.12 on p.55-57 of the text. | |
| 10/4 | No class today | |||
| 4 | 10/7 | 3.4, 4.1 | CRT, arithmetic modulo p | Exercises 3.18 and 3.20 on p.63 of the text. Exercise 4.1 on p.67 of the text. |
| 10/9 | Exam 1 in-class portion | Material through Section 3.4. Take home portion due 10/11. | ||
| 10/11 | 4.1, 4.2 | Pseudoprimes and Carmichael Numbers | Exercises 4.4 and 4.5 on p. 73 of the text. | |
| 5 | 10/14 | 5.1, Appendix B | Units and groups | |
| 10/15 | (x-hour) 5.2, class notes | Euler's function, intro to cryptography | Try out some examples with p=5 and q=7. | |
| 10/16 | class notes | More cryptography, RSA | Exercise 5.7 on p.90 of the text. | |
| 10/18 | 6.1, 6.2 | The group of units and primitive roots | Exercise 6.26(b) on p.118 of the text. Exercise 6.5 on p.101 of the text. |
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| 6 | 10/21 | 6.3-6.5 | Primitive roots for composite moduli | Read the proof of Theorem 6.7. Exercise 6.10 on p. 106 of the text. |
| 10/23 | 6.6 | Applications of primitive roots | Exercise 6.16 on p. 112 of the text. | |
| 10/25 | 7.1, 7.2 | Quadratic residues | Exercise 7.4 on p. 121 of the text. Find the set of quadratic residues mod 60. |
|
| 7 | 10/28 | 7.3 | The Legendre symbol and properties | Determine if [72] is a quadratic residue mod 41. Determine if [-2] is a quadratic residue mod 29. |
| 10/30 | Exam 2 in-class portion | Material through Section 6.6. Take home portion due 11/1. | ||
| 11/1 | 7.3 | Gauss's Lemma | Exercise 7.10 on p.129 of the text. | |
| 8 | 11/4 | 7.4 | Quadratic reciprocity | Exercise 7.12 on p.132 of the text. Read the proof of Theorem 7.11 on p.133 of the text. |
| 11/5 | (x-hour) 8.1, 8.2 | Arithmetic functions | Exercise 8.1 on p. 144 of the text. Exercise 8.5 on p. 147 of the text. |
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| 11/6 | 8.3-8.5 | The Mobius function | Exercise 8.15 on p. 156 of the text. | |
| 11/8 | 8.6, 9.1, class notes | The Dirichlet product, the Riemann Zeta function | Note: This material is not covered on the final exam. | |
| 9 | 11/11 | 11.1-11.5 | Fermat's last theorem and Pythagorean triples | Exercises on p.220 of the text. Read Section 11.8. |
| 11/13 | 10.1, 10.3 | Sums of squares I | Exercises 10.2, 10.3 on p.195 of the text. | |
| 11/15 | 10.3, 10.4 | Sums of squares II | Exercises 10.13, 10.14 on p.205 of the text. | |
| 10 | 11/18 | Wrap-up | Final Exam Study Guide |