M328K Syllabus

INTRODUCTION TO NUMBER THEORY

Prerequisite and degree relevance: required:  M341 or M325K, with a grade of at least C-.

This is a first course that emphasizes understanding and creating proofs; therefore, it must provide a transition from the problem-solving approach of calculus to the entirely rigorous approach of advanced courses such as M365C or M373K. The number of topics required for coverage has been kept modest so as to allow instructors adequate time to concentrate on developing the students theorem-proving skills.

A list of texts from which the instructor may choose is maintained in the text office.

The choice of text will determine the exact topics to be covered. The following subjects should definitely be included:

Divisibility: divisibility of integers, prime numbers and the fundamental theorem of arithmetic.

Congruences: including linear congruences, the Chinese remainder theorem, Euler's -function, and polynomial congruences, primitive roots.

The following topics may also be covered, the exact choice will depend on the text and the taste of the instructor.

Diophantine equations: (equations to be solved in integers), sums of squares, Pythagorean triples.

Number theoretic functions: the Mobius Inversion formula, estimating and partial sums n(x) of other number theoretic functions.

Approximation of real numbers by rationals: Dirichlet's theorem, continued fractions, Pell's equation, Liousville's theorem, algebraic and transcendental numbers, the transcendence of e and/or n.