M328K INTRODUCTION TO NUMBER THEORY

Prerequisite and degree relevance:

One of M311 or M341 is required, 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.

Course description:

The following subjects are included:

Divisibility:

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

Congruences:

including linear congruences, the Chinese remainder theorem, Euler’s j-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 z(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 z.