Number Theory

I. Kra

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:

- G.E. Andrews

Number Theory - T.A. Apostal

Introduction to Analytic Number Theory - P.G.L. Dirichlet with supplements by R. Dedekind

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

Lectures on Number Theory - K. Ireland and M. Rosen

A Classical Introduction to Modern Number Theory - W.J. LeVeque

Fundamentals of Number Theory - I. Niven, H.S. Zuckerman and H.L. Montgomery

An Introduction to the Theory of Numbers - G.A. Jones and J.M. Jones

Elementary Number Theory - R.A. Mollin

Fundamental NUMBER THEORY with APPLICATIONS - I. Niven and H.S. Zuckerman

An Introduction to the Theory of Numbers - C.L. Siegel

Lectures on the Geometry of Numbers - J.H. Silverman

A Friendly Introduction to Number Theory - H.M. Stark

An Introduction to the Number Theory - W.J. LeVeque

Fundamentals of Number Theory - C. Vanden Eyden

Elementary Number Theory - 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

research/scholarship/computing

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.

READING & HOMEWORK ASSIGNMENTS

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 -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) |

PROJECTS

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 or 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.