Text(s)/Materials:

Reference Text: Stark, Harold. (1970). Number Theory. Boston: MIT Press. (students use primarily as reference source; lecture notes are self-contained)

Course Description:

Survey of number theory. Intended to introduce concepts of number theory from a rigorous point of view, although not as extensive as a one-semester course for math majors.

First part of course is core material: axioms for the number system, divisibility and the GCD, factorization, Euclidean algorithm, linear diophantine equations, the multiplicative functions d(n) and s(n). Congruences, systems and the Chinese Remainder Theorem, reduced residue systems and EulerÕs f-function. Euler and Fermat theorems, primality testing, order and primitive roots. Representations of numbers in base b, periodic expansions, irrational numbers and cardinality.

Second part of course consists of optional units selected by the students. Units in recent past have included continued fractions, cryptography, quadratic reciprocity, and combinatorial game theory.

Teaching and Learning Methodology and Philosophy:

Lecture oriented. Each major topic begins with a question which leads students to investigate a phenomenon and make conjectures. Teacher then provides proofs of the theorems and techniques which can be used to solve the initial problem and related problems. Course is two-tiered, in that students can simply learn the ideas and techniques of number theory, or they can delve further into the technical details of the proofs. The course encourages students to improve their ability to write mathematical proofs, but promotes enjoyment of and interest in mathematics as its primary goal.

Student Expectations:

Students must take careful lecture notes, as not all required material is in the textbook. There are regular assignments, and students are required to keep current with these assignments. They make up a significant portion of the grade, provide practice with current material, and preview upcoming ideas. Active participation in class promotes interaction with the material and practice combining ideas from different parts of the course. This is essential to learning in this course and in mathematics in general.

Assessment Practices, Procedures, and Processes:

Assessment is very traditional. Homework assignments comprise one-third of each quarter's grade, while in-class tests and quizzes make up two-thirds. Several assignments include problems that are more difficult than usual and students can submit solutions to these for extra credit. There is a final exam which comprises 20% of the semester grade, while each quarter accounts for 40% of the final semester grade.

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