Elementary Number Theory (110.304)

Spring 2019


[Course Syllabus]

[Lecture Schedule and Assignment]


Announcements:

2/20: No office hours today (school closed). For questions regarding homework, please email me.

2/19: 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.

2/12: School opens at 10am. No class today (Read 3.2 and 3.3).

2/4: TA's office hours: Monday 11:00am-12:00pm.

1/16: The first meeting of this class is on Tuesday January 29.


 

Lecture Schedule and Assignment

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

Date Topic Sections Homework Due Date
Week 1 Feb 7
Jan 29 Introduction. Mathematical Induction 1.1 1.1: 1, 8, 10, 12, 13, 15
1.2: 6, 7
2.1: 2, 4, 5
Jan 31 Basis representation theorem
Euclid's division lemma
1.2, 2.1
Week 2 Feb 14
Feb 5 Divisibility 2.2 2.2: 2, 4, 10, 11, 12
2.3: 1(d), 4
2.4: 8, 10, 12
Feb 7 Linear Diophantine equations
Fundamental theorem of arithmetic
Permutations and combinations
2.3, 2.4, 3.1
Week 3 Feb 21
Feb 12 No class (school opens at 10am).
Read 3.2 and 3.3
  3.1: 3, 6, 7, 10, 13, 14
3.2: 3, 6
3.3: 2
3.4: 3, 5
Feb 14 Fermat's Theorem. Wilson's theorem
Generating functions
3.2 3,3, 3.4
Week 4 Feb 28
Feb 19 Basic properties of Congruences
Residue Systems
4.1, 4.2 4.1: 6
4.2: 3
5.1: 1(b), 3
5.2: 3, 4, 6, 9, 11, 15, 21, 23
Feb 21 Solving Linear Congruences
Euler's Theorem. Fermat and Wilson Theorem (using congruences).
5.1, 5.2
Week 5  
Feb 26 Chinese remainder theorem 5.3  
Feb 28 Polynomial congruences
Review
5.4
Week 6  
Mar 5 Midterm I    
Mar 7 Combinatorial study of φ(n) 6.1
Week 7  
Mar 12 Formulae for d(n) and σ(n)
Multiplicative arithmetic functions
6.2, 6.3  
Mar 14 Möbius inversion formula 6.4
Week 8  
Mar 19 No Class (spring break)    
Mar 21  
Week 9  
Mar 26 Properties of reduced residue systems 7.1  
Mar 28 Primitive Roots 7.2
Week 10  
Apr 2 Elementary properties of π(x) 8.1  
Apr 4 Tchebychev's theorem
Quadratic Residues
8.2, 9.1
Week 11  
Apr 9 Legendre Symbol. Gauss's Lemma
Quadratic reciprocity law
9.2, 9.3  
Apr 11 Applications of quadratic reciprocity law 9.4
Week 12  
Apr 16 Proof of quadratic reciprocity law
Review
9.3  
Apr 18 Midterm II  
Week 13  
Apr 23 Consecutive residues and non-residues 10.1  
Apr 25 Consecutive triples of quadratic residues 10.2
Week 14  
Apr 30 Sum of two squares 11.1  
May 2 Sum of four squares
Review
11.2