Elementary Number Theory (110.304)

Spring 2018


[Course Syllabus]

[Lecture Schedule and Assignment]


Announcements:

15/17: Course grades posted to SIS. Have a great summer!

5/17: Final exam solution posted.

5/1: Office hours for Final Exam: Tuesday May 15, 1:00-4:00pm, Krieger 412.

5/1: Reminder: The Final Exam is Wednesday, May 16, 9:00 am -12:00 noon, Maryland 104.

4/17: Office hours for Midterm II:  Wednesday April 18 2-4:30pm.

4/10: Reminder: Midterm II is on Thursday April 19.

3/6: Office hours change for Wednesday 3/7 only: 12-1pm.

2/28: Reminder: Midterm I is on Tuesday March 6.

2/20: 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.

2/7: TA's office hours: Tuesday 1-2pm. Krieger 211.

1/16: The first meeting of this class is on Tuesday January 30.


 

Lecture Schedule and Assignment

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

Date Topic Sections Homework Due Date
Week 1 Feb 8
Jan 30 Introduction. Mathematical Induction 1.1 1.1: 1, 7, 10, 11, 13, 15
1.2: 6, 7
2.1: 2, 4, 5
Feb 1 Basis representation theorem
Euclid's division lemma
1.2, 2.1
Week 2 Feb 15
Feb 6 Divisibility 2.2 2.2: 2, 10, 11, 12
2.3: 1(d), 4
2.4: 6(f), 8, 10, 12
Feb 8 Linear Diophantine equations
Fundamental theorem of arithmetic
2.3, 2.4
Week 3 Feb 22
Feb 13 Permutations and combinations
Fermat's Theorem
3.1, 3.2 3.1: 3, 6, 7, 13, 14
3.2: 3, 6
3.3: 1, 2
3.4: 3, 5
Feb 15 Wilson's theorem
Generating functions
3,3, 3.4
Week 4 Mar 1
Feb 20 Basic properties of Congruences
Residue Systems
4.1, 4.2 4.1: 1(a), 6
4.2: 3
5.1: 1(c)
5.2: 3, 4, 5, 9, 11, 15, 21, 23
Feb 22 Solving Linear Congruences
Euler's Theorem. Fermat and Wilson Theorem (using congruences).
5.1, 5.2
Week 5 Mar 8
Feb 27 Chinese remainder theorem 5.3 5.3: 2, 4, 6
5.4: 3, 4, 5, 6, 7
Mar 1 Polynomial congruences
Review
5.4
Week 6 Mar 15
Mar 6 Midterm I   6.1: 1, 4, 5, 8, 9, 10, 11, 13
Mar 8 Combinatorial study of φ(n) 6.1
Week 7 Mar 29
Mar 13 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, 11, 12
Mar 15 Möbius inversion formula 6.4
Week 8  
Mar 20 No Class (spring break)    
Mar 22  
Week 9 Apr 5
Mar 27 Properties of reduced residue systems 7.1 7.1: 6, 7
7.2: 7, 8, 9, 10, 11, 12, 13, 14
Mar 29 Primitive Roots 7.2
Week 10 Apr 12
Apr 3 Elementary properties of π(x) 8.1 8.1: 1, 2, 5, 6, 7, 10, 12, 16
8.2: 1, 7
Apr 5 Tchebychev's theorem
Quadratic Residues
8.2, 9.1
Week 11 Apr 19
Apr 10 Legendre Symbol. Gauss's Lemma
Quadratic reciprocity law
9.2, 9.3 9.1: 1
9.2: 1, 2, 3
9.3: 5, 6
9.4: 1, 3, 4, 5
Apr 12 Applications of quadratic reciprocity law 9.4
Week 12 Apr 26
(Last HW Due)
Apr 17 Proof of quadratic reciprocity law
Review
9.3 10.1: 3, 4, 5, 6, 7
Apr 19 Midterm II  
Week 13 Not Due
Apr 24 Consecutive residues and non-residues 10.1 10.2: 2
11.1
: 1, 2
11.2: 9
Apr 26 Consecutive triples of quadratic residues 10.2
Week 14  
May 1 Sum of two squares 11.1  
May 3 Sum of four squares
Review
11.2

Midterm I Solution

Midterm 2 Solution

Final Exam Solution