Difference between revisions of "Math 487: Intro to Number Theory"
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Revision as of 15:13, 3 April 2013
Contents
Catalog Information
Title
Number Theory.
(Credit Hours:Lecture Hours:Lab Hours)
(3:3:0)
Offered
F, Sp
Prerequisite
Description
Foundations; congruences; quadratic reciprocity; unique factorization, prime distribution or Diophantine equations.
Desired Learning Outcomes
This course is aimed at undergraduate mathematics majors. It is a first course in number theory, and is intended to introduce students to number theoretic problems and to different areas of number theory. Number theory has a very long history compared to some other areas of mathematics, and has many applications, especially to coding theory and cryptography.
Prerequisites
Math 371 is a prerequisite for this course. In particular, students should be familiar with the concepts of groups and rings, and they should understand constructions of quotient groups and quotient rings. By this point in their mathematical career, students should be comfortable proving theorems by themselves.
Minimal learning outcomes
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and nonexamples of the various concepts. The students should be able to demonstrate their mastery by solving nontrivial problems related to these concepts, and by proving simple (but nontrivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.
 Divisibility in the integers
 Prime numbers
 Unique factorization
 Euclid’s algorithm
 GCD and LCM
 Congruence arithmetic
 Complete and reduced residue systems
 Linear congruences
 Chinese remainder theorem
 Polynomial congruences
 Hensel’s lemma
 Quadratic residues
 Legendre and Jacobi symbols
 Quadratic reciprocity
 Primitive roots
 Existence of primitive roots
 Structure of units modulo nonprimes
 Number Theoretic Functions
 Moebius Function
 Euler phi function
 Sum of divisors function
 Big Oh notation
 Little Oh notation
 Distribution of Primes
 Definition of Pi(x)
 Estimates of Pi(x)
 Primes in arithmetic progressions
 Bertrand’s Hypothesis
 Sums of Squares
 Representations of numbers as sums of two and four squares
 Statement of Waring’s Problem
Textbooks
Possible textbooks for this course include (but are not limited to):
 Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery, An Introduction to the Theory of Numbers (5th edition), Wiley, 1991.
 George Andrews, Number Theory, Dover, 1994.
 William LeVeque, Fundamentals of Number Theory, Dover, 1996.
 Benjamin Fine and Gerhard Rosenberger, Number Theory, An Introduction via the Distribution of Primes, (available as a free Springer ebook at springerlink.com).
Additional topics
Beyond the minimal learning outcomes, instructors are free to cover additional topics. These may include (but are certainly not limited to): diophantine approximation, continued fractions, elliptic curves, cryptography, partition theory, Abel and EulerMaclaurin summation formulae. It is expected that some topics beyond the minimal learning objectives will typically be discussed.
Courses for which this course is prerequisite
This course is not a prerequisite for any other courses in the regular curriculum. Hence, this course may be the only opportunity for students to learn the topics listed here. As a result, it is important that all learning objectives be completed.