MATH 365 Elementary Number Theory I (Fall 2003)

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.
• 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.

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=a1 (mod n1),..., x=ar (mod nr) have a solution?

Chapter 5:

• (Fermat's Theorem) For a prime number p, ap-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).
• x2=-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 aΦ(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 x2=a (mod pn) have a solution?

Chapter 11:

• Find all integral solutions of x2+y2=z2.
• x4+y4=z4 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.