M328K INTRODUCTION TO NUMBER THEORY
Prerequisite and degree relevance: Required: M325K or M341, 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, Eulers 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: Dirichlets theorem, continued fractions, Pells equation, Liousvilles theorem, algebraic and transcendental numbers, the transcendence of e and/or z.