Math 487: Intro to Number Theory

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Catalog Information

Title

Number Theory.

(Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

Offered

F, Su

Prerequisite

Math 371.

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 non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.

  1. Divisibility in the integers
    • Prime numbers
    • Unique factorization
    • Euclid’s algorithm
    • GCD and LCM
  2. 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
  3. Primitive roots
    • Existence of primitive roots
    • Structure of units modulo nonprimes
  4. Number Theoretic Functions
    • Moebius Function
    • Euler phi function
    • Sum of divisors function
    • Big Oh notation
    • Little Oh notation
  5. Distribution of Primes
    • Definition of Pi(x)
    • Estimates of Pi(x)
    • Primes in arithmetic progressions
    • Bertrand’s Hypothesis
  6. 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 e-book 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 Euler-Maclaurin 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.