Elementary Number Theory (110.304)

Fall 2016


[Course Syllabus]

[Lecture Schedule and Assignment]


Announcements:

12/19: Course grades posted to SIS. Happy holidays!

12/19: Final exam solutions posted.

12/8: Extra office hours for the final exam: Thursday December15, 2-5pm.

12/8: Reminder: The Final Exam is Friday December  16, 9am-12noon in Gilman 55 (the regular classroom).

10/5: Extra office hours for Midterm I: Monday October 10, 2-4pm.

10/5: Reminder: Midterm I is on Tuesday October 11.

8/26: The first meeting is on Thursday September 1.


 

Lecture Schedule and Assignment

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

Date Topic Sections Homework Due Date
Week 1  
Sep 1 Introduction
Principle of mathematical induction
1.1 No assignment
Week 2 Sep 15
Sep 6 Basis representation theorem
Euclid's division lemma
1.2, 2.1 1.1: 2, 7, 10, 11, 13, 15
1.2: 6, 7
2.1: 2, 4, 5, 6
Sep 8 Divisibility 2,2
Week 3 Sep 22
Sep 13 Fundamental theorem of arithmetic 2.4 2.2: 2, 10, 11, 12
2.3: 1(b), 4
2.4: 6(f), 8, 10, 12
Sep 15 Linear Diophantine equations
Permutations and combinations
2.3, 3.1
Week 4 Sep 29
Sep 20 Fermat's little theorem. Wilson's theorem 3.2, 3.3 3.1: 3, 6, 7, 10, 13, 14
3.2: 3, 6
3.3: 2
3.4: 3, 5
Sep 22 Generating functions 3.4
Week 5 Oct 6
Sep 27 Basic properties of Congruences
Residue Systems
4.1, 4.2 4.1: 1(b), 6
4.2: 3
5.1: 1(c)
5.2: 3, 4, 6, 9, 11, 15, 21, 22
Sep 29 Solving Linear Congruences.
Euler's Theorem. Fermat and Wilson Theorem (using congruences).
5.1, 5.2
Week 6 Oct 13
Oct 4 Chinese remainder theorem 5.3 5.3: 2, 4, 6
5.4: 3, 4, 5, 6, 7
Oct 6 Polynomial congruences
Review
5.4
Week 7 Oct 27
Oct 11 Midterm I   6.1: 1, 4, 5, 8, 9, 10, 11, 15
Oct 13 Combinatorial study of φ(n) 6.1
Week 8 Oct 27
Oct 18 Formulae for d(n) and σ(n)
Multiplicative arithmetic functions
Möbius inversion formula
6.2, 6.3, 6.4 6.2: 2, 4, 5, 9, 15
6.3: 1
6.4: 2, 4, 5, 7, 8, 11, 12
Oct 20 No Class
(classes meet on Monday schedule)
 
Week 9 Nov 3
Oct 25 More on Möbius inversion formula
Properties of reduced residue systems
6.4, 7.1 7.1: 6, 7
7.2: 7, 8, 9, 10, 11, 12, 13, 14
Oct 27 Primitive roots 7.2
Week 10 Nov 10
Oct 1 Elementary properties of π(x) 8.1 8.1: 1, 2, 5, 6, 7, 9, 10, 12, 16
8.2: 1, 7
Nov 3 Tchebychev's theorem 8.2
Week 11 Nov 17 
Nov 8 Euler's criterion. Legendre Symbol 9.1, 9.2 9.1: 1
9.2: 1, 2, 3
9.3: 1, 2, 5, 6
9.4: 1, 3, 4, 5
Nov 10 Quadratic reciprocity law
Applications of quadratic reciprocity law
9.3, 9.4
Week 12  
Nov 15 Proof of quadratic reciprocity law. Review 9.3 No Assignment
Nov 17 Midterm II  
Week 13  
Nov 22 No Class (Thanksgiving holiday)   No Assignment
Nov 24  
Week 14 Not Due 
Nov 29 Consecutive residues and non-residues 10.1 10.1: 1, 2, 3, 4, 5, 6, 7
11.1: 1, 2
11.2: 9
Dec 1 Consecutive triples of quadratic residues 10.2
Week 15  
Dec 6 Sum of two squares 11.1 No Assignment
Dec 8 Sum of four squares. Review 11.2

Midterm I Solutions
Midterm II Solutions
Final Exam Solutions