| Week 1 |
Jan 21, 23 |
Introduction to number theory: Pythagorean triples, prime numbers, Diophantine equations. Cryptography and computation in Sage.
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Chapter 2 (optional) [Silverman] |
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| Week 2 |
Jan 28, 30 |
Greatest common divisors, the Euclidean algorithm and linear Diophantine equations.
|
Chapters 5, 6 [Silverman] |
Problem Set 1 |
| Week 3 |
Feb 4, 6 |
Introduction to proofs: direct proof, contradiction and induction.
|
TBD |
Problem Set 2 |
| Week 4 |
Feb 11, 13 |
Primes and the fundamental theorem of arithmetic.
|
Chapter 12 up to Theorem 12.1, Chapter 7 [Silverman] |
Problem Set 3 |
| Week 5 |
Feb 18, 20 |
Revisiting Pythagorean triples, introduction to congruences and solving congruence equations. Midterm 1
|
Chapter 8 [Silverman] |
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| Week 6 |
Feb 25, 27 |
More on congruences: powers, Fermat's little theorem, Euler's theorem, the phi function and the Chinese remainder theorem.
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Chapters 9, 10, 11 [Silverman] |
Problem Set 4 |
| Week 7 |
Mar 3, 5 |
(Remote lectures) Application of number theory to cryptography: RSA algorithm.
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Chapters 17, 18 [Silverman] |
Problem Set 5 |
| Week 8 |
Mar 10, 12 |
Implementation of RSA. More on cryptography and primality testing.
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Chapter 19 [Silverman] |
Problem Set 6 |
| Spring Break (Mar 16 – 20) |
| Week 9 |
Mar 24, 26 |
Squares modulo p.
|
TBD |
Problem Set 7 |
| Week 10 |
Mar 31, Apr 2 |
Quadratic reciprocity. Midterm 2
|
TBD |
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| Week 11 |
Apr 7, 9 |
Primitive roots.
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Chapters 28, 29 [Silverman] |
Problem Set 8 |
| Week 12 |
Apr 14, 16 |
Application to cryptography: Diffie-Hellman.
|
TBD |
Problem Set 9 |
| Week 13 |
Apr 21, 23 |
Other topics (possibly Pell's equations, continued fractions, irrational numbers).
|
TBD |
Problem Set 10 |