NUMBER THEORY†††††††††††† MATH 5043-01 ††††††




FALL 2002

M,W,F 12-12:50;Boyd Building 306




Instructor: ††††††††† Dr. B. Landman

Office:††††††††††††††††† 315 Boyd

Office Hours: ††††† M,W,F 1-2; Th 10:30-11, or by appointment

Phone:††††††††††††††††† 836-6489

E-mail:†††††††††† †††††

Textbook:†††††††††† Elementary Introduction to Number Theory, 3rd Edition

by Calvin T. Long;Waveland Press†††††

Prerequisite:††††††† MATH 3003


Topics:†† 1. Preliminaries (Most of Sections 1.1-1.6)

††††††††††††††† 2. Division Algorithm (Section 1.7)

†††††††††††††††††† 3. Divisibility Properties of Integers (all of Chapter 2)

†††††††††††††††††† 4. Prime Numbers (most of Chapter 3)

†††††††††††††††††† 5. Congruences (all of Chapter 4)

†††††††††††††††††† 6. Conditional Congruences (all of Chapter 5)

†††††††††††††††††† 7. Sums of Squares (Chapter 7; as time permits)

†††††††††††††††††† 8. Multiplicative Number Theory (Chapter 8; as time permits)


Learning Outcomes:††††


It is expected that the student who completes this course will have acquired:

  1. A basic knowledge of the notion of congruences, and familiarity with the methods of solving elementary congruences (L2, L4, L5, L6, L7, L10, L14)
  2. An understanding of the statements of the following theorems, and the ability to apply them: the division algorithm, the Euclidean algorithm, Lagrangeís theorem, Wilsonís theorem, the law of quadratic reciprocity, the Chinese Remainder Theorem. (L2, L4, L5, L6, L7, L10, L14)
  3. A better understanding of the basic techniques of proving mathematical statements, including the method of mathematical induction, direct proofs, proofs by contrapositive, and proofs by contradiction. The student will demonstrate this understanding by applying these techniques to prove basic number theoretical theorems and/or facts. (L2, L4)
  4. An improvement in his or her ability to read and comprehend mathematical statements and proofs. This will be measured by having the student independently read passages and proofs containing mathematical language, in the text and in assignments, and then use these statements to draw further logical conclusions or complete mathematical proofs. (L2)


  1. An understanding of the frequency of the prime numbers via the prime number theorem. Such understanding will be demonstrated via solving problems involving limit theorems concerning the set of prime numbers. (L1, L2, L4, L5, L6, L7, L10, L14)


  1. The ability to apply techniques covered in the course in a creative way to devise somewhat complex proofs ofnumber theoretical statements. This ability will be measured by having the student prove results which require the piecing together of several tools; in other words, demonstrating a higher level of familiarity and depth of understanding of the concepts than would be required in an undergraduate course. (L2, L4, L5, L6, L7, L9, L10, L14)



Grading Procedure :


Tests : There will be two 50-minute tests, and a cumulative final exam. You will be given at least ten days notice before each test. No make-up tests will be given. If you must miss one of the 50-minute tests, your grade on the corresponding portion of the final exam will be used in its place. If both tests are missed, a grade of 0 will be given for test I.

Homework (Graded): There will be approximately six sets of take-home assignments, which will be graded. You will usually have between one and two weeks to do these. Your work on these assignments is to be your own. Late papers will not be accepted. Students in this course will have more challenging homework problems than students enrolled in MATH 4043.

Homework (not graded): Homework problems (that will not be graded or collected) will be assigned regularly.These will usually be due by the next class meeting. Although not counted toward your grade, it is ESSENTIAL to your success in the course that you do the homework regularly, and on time. Part of each homework assignment is to read the corresponding material in the text.

Final Exam:There will be a comprehensive final exam.




For your grade in the course, the grade will be determined as follows


        Graded Homework 30%

        Final Exam 30%


        An overall average of88-100 earns a grade of A

An overall average of77-88†† earns a grade of B

††††††††††††††††† An overall average of66-77†† earns a grade of C

††††††††††††††††† An overall average of55-66†† earns a grade of D


Note: A stricter grading standard will be used for students enrolled in this course than in 4043.

Other:†††††††††††† It is strongly recommended that you allow at least 6-7 hours outside of class, each week, for homework and study (more would be better).

††††††††††††††††††††††† It is strongly recommended that you do not take this course if your grade in 3003 was not C or better.



Important Dates:†††††† August 19-21:††††††††††† †††††††††Drop Add/Late Registration

††††††††††††††††††††††††††††††††††† October 10, Thursday:†††††††† Last day to withdraw with grade of W

††††††††††††††††††††††††††††††††††† December 11, Wednesday:Final Exam (11:00a.m. - 1:00p.m.)