MAT 311, Spring 2001
Number Theory

I. Kra

5/2/01: The last set of homework problems need not be handed in. Solutions for these problems will be posted on 5/9.

5/1/01: The final examination will be held in Physics P128.

3/21/01: The second examination will cover the material through §8.2.

3/7/01: Tuesday (3/6) classes were cancelled because of snow.

2/13/01: Students are advised to consult the textbook's web site for useful information.

2/6/01: The first examination will cover the material through §4.2.

1/30/01: Beginning on Thursday 2/1/01, the class will meet in Math 5-127.

Course meets Tu, Th 2:20-3:40 in Math 5-127.
I. Kra office hours (Th, January 25- W, May 9) in Math 4-111: Tu 11-12, W 3-4; in Undergraduate Mathematics office: Th 11-12; and by appointment.

Text book
K.H. Rosen
ELEMENTARY NUMBER THEORY and its applications
Fourth Edition

There are many excellent undergraduate books on the subject. I have placed the following ``sample'' on reserve in the Mathematics/Physics/Astronomy Library:

  1. G.E. Andrews
    Number Theory

  2. T.A. Apostal
    Introduction to Analytic Number Theory

  3. P.G.L. Dirichlet with supplements by R. Dedekind
    Lectures on Number Theory

  4. A. Hurwitz and N. Kritikos
    Lectures on Number Theory

  5. K. Ireland and M. Rosen
    A Classical Introduction to Modern Number Theory

  6. W.J. LeVeque
    Fundamentals of Number Theory

  7. I. Niven, H.S. Zuckerman and H.L. Montgomery
    An Introduction to the Theory of Numbers

  8. G.A. Jones and J.M. Jones
    Elementary Number Theory

  9. R.A. Mollin

  10. I. Niven and H.S. Zuckerman
    An Introduction to the Theory of Numbers

  11. C.L. Siegel
    Lectures on the Geometry of Numbers

  12. J.H. Silverman
    A Friendly Introduction to Number Theory

  13. H.M. Stark
    An Introduction to the Number Theory

  14. W.J. LeVeque
    Fundamentals of Number Theory

  15. C. Vanden Eyden
    Elementary Number Theory

  16. A. Weil with collaboration of M. Rosenlicht
    Number Theory for Begginers

The books are a mixture of classical texts (for example, Dirichlet) and modern efforts (for example, LeVeque), elementary (for example, Silverman) and advanced (for example, Ireland and Rosen), algebraic (for example, Andrews) and analytic approaches (for example, Apostal). The course will concentrate on elementary algebraic number theory. More advanced analytic topics (for example, the Prime Number Theorem) require a function theoretic (complex analysis) background.

Examinations There will be a half hour pretest (on Th 2/1) that will cover very basic material from the algebra courses that are prerequisites for this course, two midterm examinations (on Tu 2/20 & Th 4/5) and a final examination (on Th 5/10). There will also be 3 un-announced twenty minute quizzes.

Projects and homework Homework is an integral part of the course. Of the 15 problem sets (of even numbered exercises), 12 are to be handed in (on first meeting (usually on a Tuesday) following the assignment); the homework grade will be based on the best 10 of these. In addition students will hand in 2 of the 15
projects. The projects with a nontrivial writing component can be used to satisfy the Mathematics Upper Division Writing Requirement.

Grading The final examination will constitute 25% of the grade; each of the midterms, 15%; the pretest 10%; the quizzes, 5% each; each of the two projects, 5%; the homework 10%. The tests and quizzes will be constructed so that at least 50% of each will consist of statements of definitions, description of key results and routine calculations. A total grade of at least 60% will be required for a C; 90% will guarantee an A.

Special needs If you have a physical, psychiatric, medical or learning disability that may impact on your ability to carry out assigned course work, you may contact the Disabled Student Services (DSS) office (Humanities 133, 632-6748/TDD). DSS will review your concerns and determine, with you, what accommodations may be necessary and appropriate. I will take their findings into account in deciding what alterations in course work you require. All information on and documentation of a disability condition should be supplied to me in writing at the earliest possible time AND is strictly confidential. Please act early, since I will not be able to make any retroactive course changes.

The following schedule may be adjusted based on students' preparation and progress.
  Topic Practice problems Hand in
Th 1/25 1.1 Numbers, sequences, and sums p14/1,5,9,25,33 2,4,26,28
Wk of 1/30 1.2 Mathematical induction p22/3,13,15 10,16,30
  1.3 The Fibonacci numbers p28/3,7,25,29 4,8,22
  1.4 Divisibility p34/3,7,11,29,58 4,6,16,30
  3.1 Prime numbers p76/3,5,9,13,29 6,10,16
Th 2/1 Half hour pretest    
Wk of 2/6 3.2 Greatest common divisors p84/13,15,31 20,22,32
  3.3 The Euclidean algorithm p94/5,7,19 20,22
  3.4 The fundamental theorem of arithmetic p104/3,11,19 6,12,20
Wk of 2/13 3.6 Linear Diophantine equations p123/3,9,21 10,20,22
  4.1 Introduction to congruences p135/5,17,25 8,20
  4.2 Linear congruences p141/1,7,15 4,10,16
Tu 2/20 First midterm (up to §4.2)    
Th 2/22 4.3 The Chinese remainder theorem p149/5,15,21 6,14,24
  4.4 Solving polynomial congruences p158/1,3,11 10,12
Wk of 2/27 4.5 Systems of linear congruences p167/1,3,5,7 2,4,12
  5.1 Divisibility tests p177/3,11,19 4,12,26
Th 3/1 First project due    
Th 3/8 6.1 Wilson's and Fermat's little theorem p202/1,3,17 14,18,34
  6.2 Pseudoprimes p213/3,9,11 6,20
Wk of 3/13 6.3 Euler's theorem p218/1,5,19 2,6,20
  7.1 The Euler $ \varphi$-function p237/1,3,19,21 4,8,22,26
  7.2 The sum and number of divisors p235/3,5,21 4,6,8
Wk of 3/27 7.4 Möbius inversion p256/3,9,17,23 6,10,24
  8.1 Character ciphers p266/3,15 14,16
  8.2 Block & stream ciphers p278/13,19 14,18
  8.3 Exponentiation ciphers p284/3,5 4,6
Tu 4/3 8.4 Public key cryptography p290/1,3,7 2,6,12
  9.1 The order of an integer and primitive roots p313/3,11,21 6,10,20
Th 4/5 Second midterm (up to §8.2)    
Wk of 4/10 9.2 Primitive roots for primes p318/7,13,15 8,12,16
  9.3 The existence of primitive roots p328/1,5,9 2,12,14
Wk of 4/17 9.4 Index arithmetic p337/3,9,21 4,10,22
Wk of 4/24 11.1 Quadratic residues and nonresidues p386/5,9,15,41 6,8,18
Tu 5/1 Second project due    
Wk of 5/1 11.2 The law of quadratic reciprocity p401/3,7,11 2,8,12
Tu 5/8 11.3 The Jacobi symbol p410/3,7,11 4,8,10
  11.4 Euler pseudoprimes p419/1,3,7,9 2,4,6
Th 5/10 Final examination (2:00 to 4:30)    

Choose one of the following projects to complete hand in by March 1.
A proof that the rational and algebraic numbers are countable and that the irrationals and transcendental numbers are not.
Programming project (from Rosen) 1.3.1.
C P E 3.1.16.
Programming project 3.3.5.
C P E 4.4.2.
Programming project 4.5.1.
Programming project 5.1.2.
A short (2 to 4 typed pages) biography of a prominent number theorist.

Choose one of the following projects to complete hand in by May 1.
A proof that $ \pi$ or $ e$ is transcendental.
Programming project 6.1.2.
C P E 7.1.4
Programming project 7.4.1.
Programming project 9.3.2.
Programming project 11.1.3.
A short (2 to 4 typed pages) paper on the mathematical contributions of a number theorist.

NOTE: For a programming project, hand in an outline of the program, the code for the program, and a reasonable amount of program output.