Elementary Number Theory (110.304)

Fall 2017


[Course Syllabus]

[Lecture Schedule and Assignment]


Announcements:

11/13: Reminder: Midterm II is Thursday, November 16.

11/13: Office hours for Midterm II: Wednesday November 15, 2:00-4:30pm, Krieger 412 (usual time but long hours).

10/4: Room change (tomorrow only): The classroom for tomorrow (October 5) will be Gilman 313, per registrar office.

10/4: Reminder: Midterm I is next Tuesday, October 10.

10/4: Extra office hours for Midterm I: Monday October 9, 2:00-3:30pm, Krieger 412.

9/26: For homework problem #14 in 3.1, here is a (3-part) help video if you use induction. Note that there are other ways without using induction.

9/19: Office hours change: for tomorrow (9/20) only, office hours will be 3:00-4:30pm.

9/18: TA's office hours: Wednesday 6:00 -- 7:00 pm. Krieger 211

9/1: Room change: Please note the new classroom for 110.304 Fall 2017 is Hodson 315, effective from first meeting on Tuesday September 5.

8/26: Please note that on the first class day August 31, classes meet according to Monday schedule. The first meeting of this class is on Tuesday September 5.


 

Lecture Schedule and Assignment

(This is tentative schedule. Check here for frequent updates.)
 

Date Topic Sections Homework Due Date
Week 0  
Aug 31 No Class (Classes meet on Monday schedule)   No assignment
Week 1 Sep 14
Sep 5 Introduction. Mathematical Induction 1.1 1.1: 2, 7, 10, 12, 13, 15
1.2: 6, 7
2.1: 2, 4, 5, 7
Sep 7 Basis representation theorem
Euclid's division lemma
1.2, 2.1
Week 2 Sep 21
Sep 12 Divisibility 2.2 2.2: 2, 10, 11, 12
2.3: 1(b), 4
2.4: 6(f), 8, 10, 12
Sep 14 Linear Diophantine equations
Fundamental theorem of arithmetic
2.3, 2.4
Week 3 Sep 28
Sep 19 Permutations and combinations 2.4, 3.1 3.1: 3, 6, 7, 10, 13, 14
3.2: 3, 6
3.3: 2
3.4: 3, 5
Sep 21 Fermat's little theorem. Wilson's theorem
Generating functions
3.2, 3,3, 3.4
Week 4 Oct 5 
Sep 26 Basic properties of Congruences
Residue Systems
4.1, 4.2 4.1: 1(c), 6
4.2: 3
5.1: 1(b)
5.2: 3, 4, 6, 9, 11, 15, 21, 22
Sep 28 Solving Linear Congruences
Euler's Theorem. Fermat and Wilson Theorem (using congruences).
5.1, 5.2
Week 5 Oct 12
Oct 3 Chinese remainder theorem 5.3 5.3: 2, 4, 6
5.4: 3, 4, 5, 6, 7
Oct 5 Polynomial congruences
Review
5.4
Week 6 Oct 19
Oct 10 Midterm I   6.1: 1, 4, 5, 8, 9, 10, 11, 15
Oct 12 Combinatorial study of φ(n) 6.1
Week 7 Oct 26
Oct 17 Formulae for d(n) and σ(n)
Multiplicative arithmetic functions
6.2, 6.3 6.2: 2, 4, 5, 9, 15
6.3: 1
6.4: 2, 4, 5, 7, 8, 11, 12
Oct 19 Möbius inversion formula 6.4
Week 8 Nov 2
Oct 24 Properties of reduced residue systems
Primitive roots
7.1, 7.2 7.1: 6, 7
7.2: 7, 8, 9, 10, 11, 12, 13, 14
Oct 26 Elementary properties of π(x) 8.1
Week 9 Nov 9
Oct 31 Tchebychev's theorem 8.2 8.1: 1, 2, 3, 5, 6, 7, 9, 10, 18
8.2: 1
Nov 2 Euler's criterion. Legendre Symbol 9.1, 9.2
Week 10 Nov 16
Nov 7 Gauss's Lemma. Quadratic reciprocity law 9.3 9.1: 1
9.2: 1, 2, 3
9.3: 1, 2, 5, 6
9.4: 1, 3, 4, 5
Nov 9 Quadratic reciprocity law
Applications of quadratic reciprocity law
9.4
Week 11 Nov 30
Nov 14 Consecutive residues and non-residues Review 10.1 10.1: 2, 4, 5, 6, 7
Nov 16 Midterm II  
Week 12  
Nov 21 No Class (Thanksgiving holiday)    
Nov 23  
Week 13  
Nov 28 Consecutive triples of quadratic residues 10.2  
Nov 30 Sum of two squares 11.1
Week 14  
Dec 5 Sum of four squares 11.2  
Dec 7 Review  

Midterm I solution