# MAT 447 Number Theory

This is a sample syllabus only. Ask your instructor for the official syllabus for your course.

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### Course Description

Divisibility, congruences, prime number theory, Diophantine equations and other selected topics from elementary number theory.

MAT 447 meets for three hours of lecture per week.

### Prerequisites

MAT 271 with a grade of "C" or better.

### Objectives

After completing MAT 447 the student should be able to solve simple problems, do simple proofs and state basic definitions and theorems involving:

• Divisibility, congruence and combinatorics
• The Euclidean Algorithm, the Chinese Remainder Theorem and properties of primes
• Important congruence relations, Fermat's Little Theorem, Lagrange's Theorem, Wilson's Theorem, etc.
• Important arithmetic functions, multiplicativity, Mobius Inversion
• Primitive roots and quadratic reciprocity
• Diophantine Equations and Fermat's Last Theorem

### Expected outcomes

Students should be able to demonstrate through written assignments, tests, and/or oral presentations, that they have achieved the objectives of MAT 447.

### Method of Evaluating Outcomes

Evaluations are based on homework, class participation, short tests and scheduled examinations covering students' understanding of topics covered in MAT 447.

### Text

Introduction to Number Theory, by Peter Schumer. PWS Publishing Company, 1995.

• Chapter 1 - Background
• Chapter 2 - Congruences and Prime Factorization
• 2.1 The Euclidean Algorithm and Some Consequences
• 2.2 Congruence Equations and the Chinese Remainder Theorem
• 2.3 Primes and the Fundamental Theorem of Arithmetic
• 2.4 Introduction to Primality Testing and Factoring
• 2.5 Some Important Congruence Relations
• 2.6 general Polynomial Congruences: Hensel's Lemma
• Chapter 3 - Arithmetic Functions
• 3.1 Examples of Arithmetic Functions
• 3.2 Multiplicativity
• 3.3 Mobius Inversion and Some Consequences
• 3.4 Perfect Numbers and Amicable Pairs
• Chapter 4 - Primitive Roots and Quadratic Reciprocity
• 4.1 primitive Roots
• 4.2 Quadratic and nth Power Residues
• 4.3 The Legendre Symbol and Gauss's Lemma
• 4.4 The Law of Quadratic Reciprocity and Extensions
• Chapter 5 - Sums of Squares
• 5.1 Fundamentals of Diophantine Equations
• 5.2 Sums of Two Squares
• Chapter 8 - Introduction to Analytic Number Theory
• 8.1 The Infinitude of Primes and the Zeta Function

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in MAT 447. The instructor determines the relative weights of these factors.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates and Make-Up Work

Due dates and policy regarding make-up work are set by the instructor.

### Schedule of Examinations

The instructor sets all test dates except the date of the final exam. The final exam is given at the date and time announced in the Schedule of Classes.