MATH 365 Elementary Number Theory I (Fall 2003)
2003 - 04 Fall
MATH 365: Elementary Number Theory I
The positive integers stand there, a continual and inevitable
challenge to the curiousity of every healthy mind. --- G. H. Hardy
- Announcements:
- Make-up Exam: Let's agree on 19 January 2004 Monday 10:00 am for the time being. If there is a conflict we can postpone it to Monday afternoon or Tuesday. Let me know before 14 January 2004 Wednesday.
- The final exam will be on 14 January 2004 Wednesday 5:00 pm as announced.
- The second exam will be on 3 December 2003. Section 9.3 is included.
- The Fall issue of
Matematik D�nyasi is out now.
- The first exam will be on 23 October 2003 Thursday at 17:40.
- Please check this page for future announcements.
- Lectures:
Monday 15:40-17:30 (M-103) Thursday 13:40-14:30 (M-104)
- Textbook. David Burton, Elementary Number Theory, McGraw-Hill, Fourth Edition, 1998.
Office Hours:
Monday 12:40 - 13:30
Thursday 10:40 - 11:30
Exams:
Exam 1:
Date and Time: 23 October 2003 Thursday 17:40
Place: B-14
Contents: Chapters 2, 3, 4, 5 and section 6.1. (The proof of Bonse's Inequality is also included.)
Exam 2:
Date and Time: 3 December 2003 Wednesday 17:40
Place: [A-I] M-103, [K-Z] M-104
Contents: 6.2-6.3, 7.1-7.4, 8.1-8.3, 9.1-9.3.
Final:
Date and Time: 14 January 2004 17:00
Place: [A-J] M-102, [K-Z] M-13
Contents: Chapters 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13.
Grading:
Exam 1 - 30 points
Exam 2 - 30 points
Final - 40 points
Some facts, questions and open problems that we will study in this course:
Chapter 2:
- Given integers a, b, c; when does ax+by=c have an integral solution?
Chapter 3:
- Are there infinitely many primes?
- Are there infinitely many primes of the form 4n+3?
- Are there infinitely many twin primes?
- Is every even integer greater than 2 a sum of two primes?
Chapter 4:
- When does ax=b (mod n) have a solution?
- When does a system x=a_{1} (mod n_{1}),...,
x=a_{r} (mod n_{r}) have a solution?
Chapter 5:
- (Fermat's Theorem) For a prime number p, a^{p-1}=1 (mod p)
if a is not a multiple of p.
- (Wilson's Theorem) For a prime number p, (p-1)!=-1 (mod p).
- x^{2}=-1 (mod p) has a solution iff p=1 (mod 4).
Chapter 6:
- Some properties of certain arithmetic functions will be discussed.
Chapter 7:
- Properties of Euler's Φ-function.
- For example, if gcd(a,n)=1, then a^{Φ(a)}=1 (mod n).
Chapter 8:
- 2, 4, powers of odd primes and doubles of powers of odd primes are the only integers which have primitive roots.
Chapter 9:
- Let p be an odd prime and gcd(a,p)=1. When does x^{2}=a
(mod p^{n}) have a solution?
Chapter 11:
- Find all integral solutions of x^{2}+y^{2}=z^{2}.
- x^{4}+y^{4}=z^{4} has no solution in
positive integers.
Chapter 12:
- Which numbers are sum (or difference) of two squares?
Sum of three squares? Sum of four squares?
Chapter 13:
- 1, 1, 2, 3, 5, 8, 13, .....
Last updated on 2 January 2004.