Elementary Number Theory (110.304)

Fall 2019


[Course Syllabus]

[Lecture Schedule and Assignment]


Announcements:

11/5: Reminder: Midterm II is next Thursday, November 14.

10/1: Extra office hours for Midterm I: Monday October 8, 2:00-3:30pm, Krieger 412.

10/1: Reminder: Midterm I is next Tuesday, October 9.

9/30: TA's new hours in Math Helproom will be Thursday 7-9pm. Also you can usually find him in his office (Krieger 201) Tuesday 3-4pm.

9/23: 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. 

9/6: TA's office hours: Thursday 4-5pm, Krieger 201. TA also will be in Math Helproom Tuesday 3-5pm.

8/16: Please note that on the first class day August 29, classes meet according to Monday schedule. The first meeting of this class is on Tuesday September 3.


 

Lecture Schedule and Assignment

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

Date Topic Sections Homework Due Date
Week 0  
Aug 29 No Class (Classes meet on Monday schedule)   No assignment
Week 1 Sep 12
Sep 3 Introduction. Mathematical Induction 1.1 1.1: 1, 7, 10, 11, 13, 15
1.2: 6, 7
2.1: 2, 4, 5
Sep 5 Basis representation theorem
Euclid's division lemma
1.2, 2.1
Week 2 Sep 19
Sep 10 Divisibility
Linear Diophantine equations
2.2, 2.3 2.2: 3, 10, 11, 12
2.3: 1(f), 4
2.4: 6(f), 8, 10, 12
Sep 12 Fundamental theorem of arithmetic 2.4
Week 3 Sep 26
Sep 17 Permutations and combinations
Fermat's little theorem
3.1, 3.2 3.1: 3, 6, 7, 10, 14
3.2: 3, 6
3.3: 2
3.4: 3, 5
Sep 19 Wilson's theorem. Generating functions 3.3, 3.4
Week 4 Oct 3
Sep 24 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, 6, 9, 11, 15, 21, 22
Sep 26 Solving linear congruences
Euler's Theorem. Fermat and Wilson Theorem (using congruences).
5.1, 5.2
Week 5 Oct 10
Oct 1 Chinese remainder theorem 5.3 5.3: 4, 5, 6
5.4: 1(c), 3, 4, 5, 6
Oct 3 Polynomial congruences.
Review
5.4
Week 6 Oct 17
Oct 8 Midterm I   6.1: 1, 4, 5, 8, 9, 10, 11
6.2: 2, 4, 5
Oct 10 Combinatorial study of φ(n)
Formulae for d(n) and σ(n)
6.1, 6.2
Week 7 Oct 24
Oct 15 Multiplicative arithmetic functions
Möbius inversion formula
6.3, 6.4 6.2: 9, 15
6.3: 1
6.4: 2, 4, 5, 6, 7, 11, 12
Oct 17 Properties of reduced residue systems 7.1
Week 8 Oct 31
Oct 22 Primitive roots 7.2 7.1: 6, 7
7.2: 7, 8, 9, 10, 11, 12, 13, 14
Oct 24 Elementary properties of π(x) 8.1
Week 9 Nov 7
Oct 29 Tchebychev's theorem 8.2 8.1: 1, 2, 5, 6, 7, 9, 10, 16, 18
8.2: 1
Oct 31 Euler's criterion. Legendre Symbol 9.1, 9.2
Week 10 Nov 14
Nov 5 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
Nov 7 Applications of quadratic reciprocity law 9.4
Week 11 Nov 21
(Last HW due)
Nov 12 Consecutive residues and non-residues Review 10.1 10.1: 2, 4, 5, 6, 7
Nov 14 Midterm II  
Week 12  
Nov 19 Consecutive triples of quadratic residues 10.2  
Nov 21 Sum of two squares 11.1
Week 13  
Nov 26 No Class (Thanksgiving holiday)    
Nov 28  
Week 14  
Dec 3 Sum of four squares 11.2  
Dec 5 Review  

Midterm I solution