Weekly Schedule
| Week | Dates | Topics | Reading | Relevant Notes | Problem Sets |
|---|---|---|---|---|---|
| Week 1 | Jan 20, 22 | Introduction to number theory: Pythagorean triples, prime numbers, Diophantine equations. Cryptography and computation in Sage. | Chapter 2 (optional) [Silverman] |
Divisibility and GCD Euclidean Algorithm |
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| Week 2 | Jan 27, 29 | Greatest common divisors, the Euclidean algorithm and linear Diophantine equations. | Chapters 5, 6 [Silverman] | Exploratory Worksheet | Problem Set 1 (Due Jan 30) |
| Week 3 | Feb 3, 5 | Introduction to proofs: direct proof, contradiction and induction. |
Supplementary References: • Book of Proof (In depth reference for proofs and logic) • Discrete Math: Logic Chapter (quick reference) |
Intro to Proofs and Logic Worksheet | Problem Set 2 (Due Feb 6) |
| Week 4 | Feb 10, 12 | The fundamental theorem of arithmetic. Applications: irrationality, diophantine equations. | Chapter 12 up to Theorem 12.1, Chapter 7 [Silverman] | Problem Set 3 (Due Feb 13) | |
| Week 5 | Feb 17, 19 | Introduction to congruences and solving congruence equations. Midterm 1 | Chapter 8, 9 [Silverman] |
Introduction to Modular Arithmetic Units, Order and Fermat's Little Theorem |
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| Week 6 | Feb 24, 26 | More on congruences: powers, Fermat's little theorem, Euler's theorem, the phi function and the Chinese remainder theorem. | Chapters 9, 10, 11 [Silverman] | Euler's Theorem and the Chinese Remainder Theorem | Problem Set 4 (Due Feb 27) |
| Week 7 | Mar 3, 5 | (Remote lectures) Application of number theory to cryptography: RSA algorithm. | Chapters 17, 18 [Silverman] | Cryptography and RSA | Problem Set 5 (Due Mar 6) |
| Week 8 | Mar 10, 12 | Primality testing and pseudoprimes. | Chapter 19 [Silverman] | Problem Set 6 (Due Mar 25) | |
| Spring Break (Mar 16 – 20) | |||||
| Week 9 | Mar 24, 26 | Squares modulo p. | TBD | Problem Set 7 (Due Mar 27) | |
| Week 10 | Mar 31, Apr 2 | Quadratic reciprocity. Midterm 2 | TBD | — | |
| Week 11 | Apr 7, 9 | Primitive roots. | Chapters 28, 29 [Silverman] | Problem Set 8 (Due Apr 10) | |
| Week 12 | Apr 14, 16 | Application to cryptography: Diffie-Hellman. | TBD | Problem Set 9 (Due Apr 17) | |
| Week 13 | Apr 21, 23 | Other topics (possibly Pell's equations, continued fractions, irrational numbers). | TBD | Problem Set 10 (Due Apr 24) | |