Elementary Number Theory (110.304)

Fall 2017


[Course Syllabus]

[Lecture Schedule and Assignment]


Announcements:

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  
Sep 26 Basic properties of Congruences
Residue Systems
4.1, 4.2  
Sep 28 Solving Linear Congruences. 5.1
Week 5  
Oct 3 Euler's Theorem. Fermat and Wilson Theorem (using congruences). 5.2  
Oct 5 Chinese remainder theorem
Review
5.3
Week 6  
Oct 10 Midterm I    
Oct 12 Polynomial congruences 5.4
Week 7  
Oct 17 Combinatorial study of φ(n) 6.1  
Oct 19 Formulae for d(n) and σ(n)
Multiplicative arithmetic functions
Möbius inversion formula
6.2, 6.3, 6.4
Week 8  
Oct 24 More on Möbius inversion formula
Properties of reduced residue systems
6.4, 7.1  
Oct 26 Primitive roots 7.2
Week 9  
Oct 31 Elementary properties of π(x) 8.1  
Nov 2 Tchebychev's theorem 8.2
Week 10  
Nov 7 Euler's criterion. Legendre Symbol 9.1, 9.2  
Nov 9 Quadratic reciprocity law
Applications of quadratic reciprocity law
9.3, 9.4
Week 11  
Nov 14 Proof of quadratic reciprocity law. Review 9.3  
Nov 16 Midterm II  
Week 12  
Nov 21 No Class (Thanksgiving holiday)    
Nov 23  
Week 13  
Nov 28 Consecutive residues and non-residues 10.1  
Nov 30 Consecutive triples of quadratic residues 10.2
Week 14  
Dec 5 Sum of two squares 11.1  
Dec 7 Sum of four squares. Review 11.2