| 1 |
Aug 17 |
Introduction; Induction; the sum 1 + ... + n |
1-1 |
HW1 assigned. |
| 2 |
Aug 22 |
Basis representation |
1-2 |
|
| 3 |
Aug 24 |
Euclid's Division Lemma; GCDs; Integer Combinations |
2-1 |
- |
| 4 |
Aug 29 |
Extended Euclidean Algorithm; primes |
2-2 |
- |
| 5 |
Aug 31 |
Linear Diophantine Eqns; Fundamental Theorem of Arithmetic |
2-3,2-4 |
HW1 due; HW2 assigned. |
| 6 |
Sep 5 |
Permutations and Combinations |
3.1,3.2 |
- |
| 7 |
Sep 7 |
Fermat's little theorem; Wilson's theorem |
3.2,3.3 |
- |
| 8 |
Sep 12 |
Congruences; Complete and Reduced Residue Systems |
4.1,4.2 |
- |
| 9 |
Sep 14 |
Riffling (congruence application); Linear congruence relations |
4.3,5.1 |
HW2 due; HW3 assigned. |
| 10 |
Sep 19 |
Euler's Theorem; Fermat and Wilson II |
5.2 |
- |
| 11 |
Sep 21 |
Systems of congruences; Chinese remainder theorem; Polynomial congruences |
5.3,5.4 |
- |
| 12 |
Sep 26 |
Combinatorial study of Euler's phi function |
6.1 |
- |
| 13 |
Sep 28 |
Multiplicative functions; Dirichlet product |
6.2-6.4 |
HW3 due; HW4 assigned. |
| 14 |
Oct 3 |
Mobius inversion |
6.4 |
- |
| 15 |
Oct 5 |
Midterm Exam (coverage up to and including Thurs Sept 28) |
- |
Midterm (pdf) |
| 16 |
Oct 10 |
Orders modulo m and primitive roots |
7.1 |
- |
| 17 |
Oct 12 |
Existence of primitive roots modulo a prime; prime counting function |
7.2,8.1 |
- |
| 18 |
Oct 17 |
Divergence of Sum 1/p, Chebychev's Thm |
8.2 |
HW4 due. |
| 19 |
Oct 19 |
Bertrand's postulate |
8.2 |
HW5 assigned. |
| 20 |
Oct 24 |
Open problems about primes; Euler's Criterion; Legendre Symbol |
8.2,9.1,9.2 |
- |
| 21 |
Oct 26 |
Gauss's Lemma; Legendre symbols (-1/p) and (2/p) |
9.3 |
- |
| 22 |
Oct 31 |
Quadratic Reciprocity Law I |
9.3 |
- |
| 23 |
Nov 2 |
Quadratic Reciprocity Law II |
9.3,9.4 |
HW5 due; HW6 assigned. |
| 24 |
Nov 7 |
Consecutive quadratic residue pairs |
10.1 |
- |
| 25 |
Nov 9 |
Consecutive quadratic residue triples I |
10.2 |
- |
| 26 |
Nov 14 |
Consecutive quadratic resuide triples II |
10.2 |
- |
| 27 |
Nov 16 |
Numbers representable as the sums of two squares |
11.1 |
HW6 due; HW7 assigned. |
| 28 |
Nov 28 |
Partitions |
12.1 |
- |
| 29 |
Nov 30 |
Generating functions; Partition identities |
13.1 |
- |
| 30 |
Dec 5 |
Geometric number theory |
15 |
- |
| - |
Dec 13 |
Final Exam: Wed Dec 13 11am to 1pm |
- |
- |