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.