Elementary Number Theory (110.304)

Spring 2019


[Course Syllabus]

[Lecture Schedule and Assignment]


Announcements:

5/17: Course grades posted to SIS. Have a great summer!

5/17: Final exam solution posted.

5/1: Office hours for Final Exam: Wednesday May 15, 3:30-5:30pm, Krieger 412.

5/1: Reminder: The Final Exam is Thursday, May 16, 2:00--5:00pm.

4/15: Office hours for Midterm II:  Wednesday April 17 2-3:30pm.

4/15: Reminder: Midterm II is on Thursday April 18.

2/28: Extra office hours for Midterm I: Monday March 4, 2-3:30pm.

2/26: Reminder: Midterm I is next Tuesday March 5.

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 Mar 7
Feb 26 Chinese remainder theorem
Polynomial congruences
5.3 5.3: 2, 5, 6
5.4: 1(a), 3, 4, 5, 6
Feb 28 Review 5.4
Week 6 Mar 14
Mar 5 Midterm I   6.1: 1, 2, 4, 5, 8, 9, 10, 11, 13
Mar 7 Combinatorial study of φ(n) 6.1
Week 7 Mar 28
Mar 12 Formulae for d(n) and σ(n)
Multiplicative arithmetic functions
6.2, 6.3 6.2: 2, 5, 9, 15
6.3: 1
6.4: 2, 4, 5, 7, 10, 11, 12
Mar 14 Möbius inversion formula 6.4
Week 8  
Mar 19 No Class (spring break)    
Mar 21  
Week 9 Apr 4
Mar 26 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
Mar 28 Elementary properties of π(x) 8.1
Week 10 Apr 11
Apr 2 Tchebychev's theorem 8.2 8.1: 1, 2, 3, 5, 6, 7, 10, 13, 16
8.2: 1
Apr 4 Quadratic Residues. Legendre Symbol 9.1, 9.2
Week 11 Apr 18
Apr 9 Gauss's Lemma
Quadratic reciprocity law
9.3 9.1: 1
9.2: 1, 2, 3
9.3: 5, 6
9.4: 1, 3, 4, 5
Apr 11 Applications of quadratic reciprocity law 9.3, 9.4
Week 12 Apr 25
(Last HW Due)
Apr 16 Consecutive residues and non-residues
Review
10.1 10.1: 3, 4, 5, 6, 7
Apr 18 Midterm II  
Week 13 Not Due
Apr 23 Consecutive triples of quadratic residues 10.2 10.2: 2
11.1
: 1
11.2: 9
Apr 25 Sum of two squares 11.1
Week 14  
Apr 30 Sum of four squares 11.2  
May 2 Review  

Midterm I solution

Midterm II solution

Final Exam solution