Elementary Number Theory (110.304)

Fall 2018


[Course Syllabus]

[Lecture Schedule and Assignment]


Announcements:

10/2: Reminder: Midterm I is next Tuesday, October 9.

10/2: Extra office hours for Midterm I: Monday October 8, 3:00-4:30pm, Krieger 412.

9/24: 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. 

9/17: Office hour change (this week only): This week's office hours will be Tuesday 9/18 (tomorrow) 2:00-3:30pm. There will be no office hours this Wednesday.

9/8: TA's office hour: Thursday 4:00-5:00pm in Krieger 211. Please note that the TA will also be in math Helproom Wednesday 5:00-7:00pm. 

8/26: Please note that on the first class day August 30, classes meet according to Monday schedule. The first meeting of this class is on Tuesday September 4.


 

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 13
Sep 4 Introduction. Mathematical Induction 1.1 1.1: 1, 7, 10, 12, 13, 15
1.2: 6, 7
2.1: 2, 3, 4, 5
Sep 6 Basis representation theorem
Euclid's division lemma
1.2, 2.1
Week 2 Sep 20
Sep 11 Divisibility
Linear Diophantine equations
2.2, 2.3 2.2: 3, 10, 11, 12
2.3: 1(d), 4
2.4: 6(f), 8, 10, 12
Sep 13 Fundamental theorem of arithmetic
Permutations and combinations
2.4, 3.1
Week 3 Sep 27
Sep 18 Fermat's little theorem. Wilson's theorem 3.2, 3.3 3.1: 3, 6, 7, 13, 14
3.2: 3, 6
3.3: 2
3.4: 3, 5
Sep 20 Generating functions
Basic properties of congruences
3.4, 4.1
Week 4 Oct 4
Sep 25 Residue systems.
Solving linear congruences
4.2, 5.1 4.1: 1(b), 6
4.2: 3
5.1: 1(c)
5.2: 3, 4, 5, 9, 11, 15, 21, 22
Sep 27 Euler's Theorem. Fermat and Wilson Theorem (using congruences). 5.2
Week 5 Oct 11
Oct 2 Chinese remainder theorem 5.3 5.3: 4, 5, 6
5.4: 1(1), 3, 4, 5, 6
Oct 4 Polynomial congruences.
Properties of  φ(n)
Review
5.4, 6.1
Week 6 Oct 18
Oct 9 Midterm I   6.1: 1, 4, 5, 8, 9, 10, 11
6.2: 2, 4, 5
Oct 11 Combinatorial study of φ(n)
Formulae for d(n) and σ(n)
Multiplicative arithmetic functions
6.1, 6.2, 6.3
Week 7  
Oct 16 Möbius inversion formula 6.4  
Oct 18 Properties of reduced residue systems 7.1
Week 8  
Oct 23 Primitive roots 7.2  
Oct 25 Elementary properties of π(x) 8.1
Week 9  
Oct 30 Tchebychev's theorem 8.2  
Nov 1 Euler's criterion. Legendre Symbol 9.1, 9.2
Week 10  
Nov 6 Gauss's Lemma. Quadratic reciprocity law 9.3  
Nov 8 Quadratic reciprocity law
Applications of quadratic reciprocity law
9.4
Week 11  
Nov 13 Consecutive residues and non-residues Review 10.1  
Nov 15 Midterm II  
Week 12  
Nov 20 No Class (Thanksgiving holiday)    
Nov 22  
Week 13  
Nov 27 Consecutive triples of quadratic residues 10.2  
Nov 29 Sum of two squares 11.1
Week 14  
Dec 4 Sum of four squares 11.2  
Dec 6 Review  

Midterm I Solution