# Mathematics (MATH) 409

## Number Theory (Revision 2)

Delivery Mode: Individualized study online

Credits: 3

Area of Study: Science

Prerequisite: MATH 265, MATH 266, MATH 270, MATH 271 and MATH 309, or equivalent courses from another university.

Centre: Centre for Science

MATH 409 is not available for challenge.

## Overview

Mathematics 409 is a course in elementary number theory, the branch of mathematics concerned with the properties of numbers, including, but not limited to, divisibility, prime numbers, modular arithmetic, quadratic congruences, Pythagorean triples, and the theorems, conjectures, definitions and lemmas that permit exploration of these topics.

## Outline

MATH 409 comprises the following 11 units.

• Unit 0: Introduction to Mathematics 409
• Unit 1: Division and Linear Diophantine Equations
• Unit 2: Introduction to the Prime Numbers
• Unit 3: Modularity
• Unit 4: Fermat’s Little Theorem and Euler’s Theorem
• Unit 5: Introduction to Cryptography
• Unit 6: Primitive Roots
• Unit 8: Pythagorean Triples and Sums of Squares
• Unit 9: Rational Approximation and Pell Equations
• Unit 10: Finding Prime Numbers

## Learning Outcomes

Upon successful completion of this course, you should be able to

• demonstrate a foundational understanding of number theory, including the definitions, conjectures, and theorems that permit exploration of topics in the field.
• define and determine whether a number is a prime.
• state and apply the fundamental theorem of arithmetic.
• work with numbers and polynomials modulo a prime, linear congruences, and systems of linear congruences, including their solution via the Chinese remainder theorem.
• define the order of a number relative a prime and be able to restate and apply Fermat’s little theorem, Euler’s theorem and Wilson’s theorem.
• state and apply Lagrange’s theorem, define a primitive root of a prime p, and derive properties of the Euler φ-function.
• define quadratic residues and non-residues for primes p, determine whether a number is a quadratic residue for a prime, and state and prove the law of quadratic reciprocity.
• define Pythagorean triples and primitive Pythagorean triples and derive their properties.
• use number theory to design an effective encryption system through an exploration of the RSA public key cryptosystem.

## Evaluation

To receive credit for MATH 409, a student must achieve a course composite grade of at least a C— (60 percent). A student must achieve combined grade of at least 50 per cent on the midterm and final examinations.

The course notebook consists of 10 units (see “Special Course Features,” below) and accounts for 70 percent of the final grade; the midterm examination for 10 percent, and the final examination for 20 percent.

The weighting of the composite grade is as follows:

Activity Weighting
Course notebook (10 Units) 70%
Midterm Examination 10%
Final Exam 20%
Total 100%

## Course Materials

### Textbooks

Marshall, David C., Edward Odell, and Michael Starbird. Number Theory Through Inquiry, 1st ed. Washington, DC: Mathematical Association of America, Inc., 2007.

### Other Materials

The course materials also include a study guide and a student manual.

### Special Course Features

The teaching method employed in Mathematics 409 is a version (adjusted for distance education) of what is known as the Moore method, developed by Robert L. Moore, who taught mathematics at the University of Texas from about 1920 to 1969. Traditionally, Moore-method instruction takes place in a small classroom situation with a maximum of about twenty students. The method is based on the realization that students gain a deeper understanding of a topic when they explore it for themselves.

As a record of this exploration, each student is expected to keep a course notebook, divided into chapters, in which they write out solutions to exercises, proofs of theorems, answers to questions, general ideas on the course material, and other observations relevant to the course. This notebook is the student's record of progress in their exploration of number theory. When the student completes each chapter in the textbook, they will send copies of the corresponding pages of the course notebook to the tutor for grading.

Course notebook chapters will be graded on the basis of the solutions to exercises, answers to questions and the proofs they contain. A student cannot receive credit for any exercise or proof that they have seen before, taken from a book or the Internet, or received assistance on from any person. In other words, students will be graded only on work they have completed on their own, without assistance.

We do not expect that any student will be able to answer every question, work every exercise or prove every theorem. Thus, the notebook chapters will be graded on the basis of what the student accomplished and on their improvement over the length of the course. In other words, a student's grade on the course notebook will depend on the quality of the work shown, rather than on the quantity. In this way, the record of the grades for each unit will reflect the student's growing understanding both of number theory and of how to think creatively in mathematics.

Athabasca University reserves the right to amend course outlines occasionally and without notice. Courses offered by other delivery methods may vary from their individualized-study counterparts.

Opened in Revision 2, April 25, 2015.