# Number Theory

### Number Theory Syllabus

• Course Code: Math 343

Transcript:

Yes. Your transcript will come from the records office at United States University. They are regionally accredited and award semester credits.

Credits: 4 Semester

Transfer: 4 year degree applicable

Enrollment Schedule:

Enroll any day of the year, and start that same day. Students have five months of access, plus a 30 day extension at the end if needed. Students can finish the self-paced courses as soon as they are able. Most students finish the lower level courses in 4 - 8 weeks. The upper level math classes, such as Calculus and above, usually take students 3-4 months. (Note: The 30-day extension cannot take your total course time six months beyond the date of enrollment. At the end of the six months, we must post a grade with the university.)

Required Textbook:

Number Theory (Dover Books on Mathematics)

There is only one edition to this paperback, and it can be purchased on Amazon.
- George E. Andrews
- Textbook ISBN-10: 0-486-68252-8
- Textbook ISBN-13: 0-486-68252-5

Proctored Final: Yes

#### Description

An introduction to the principles and concepts of Number Theory. Topics include distribution of primes, representations of integers, Fibonacci numbers, divisibility, Euclidean algorithm, fundamental theorem of arithmetic, number-theoretic functions, Diophantine equations, congruence, primitive roots, the Chinese remainder theorem, quadratic residues, and elementary partition theory.
Prerequisite: Methods of Proof and Linear Algebra with a grade of C or better.

Note: This course is proof based. All homework assignments, exams and the final are graded by the instructor.

#### Learning Outcomes

At the conclusion of this course, students should be able to:

1. Apply divisibility properties and the Fundamental Theorem of Arithmetic.
2. Use the Euclidean algorithm to find the greatest common factor of two integers.
3. Find integral solutions to specified linear Diophantine equations.
4. Apply the Chinese remainder theorem.
5. Solve systems of linear congruencies.
6. Understand the basic properties of integers, including divisibility, primality, congruence, number theoretic functions, and their applications.
7. Apply Fermat's Little theorem to prove relations involving prime numbers.
8. Use Wilson's theorem to solve congruencies involving factorials.
9. Apply the Mobius Inversion formula.
10. Understand current unsolved problems about primes and their importance to the field of mathematics and other related fields.

#### Methods of Evaluation:

Homework 15%
Exams (4) 60%
Final 25%
(You must get at least 60% on this final in order to pass the class with a C or better.)

Homework:
Homework assignments are essential in a mathematics course. It is not possible to master the course without a considerable amount of time being devoted to studying the concepts and using the concepts to solve problems. Each lesson contains a set of homework problems.

Exams (4):
The exams are designed to cover a broader area of the text and test your understanding of the material. Typically, exams will cover two chapters of the text.

#### Proctored Final: 25%

The final exam is a comprehensive final covering all of the chapters covered in the course.

Authorized materials for the exam:

• Writing utensil(s) and eraser(s).
• Scratch Paper (must be turned in to the proctor after the exam)
• Graph paper.

Additional items may be used for students in the following classes:

• Linear Algebra students: - Graphing calculator.
• Trigonometry & Precalculus Students - Trig & Precal students may use a predefined set of notes. Please print the set of formula notes the final. No other notes may be added to these pages. If any notes are added, the student will receive an F for cheating.
• Statistics Students - Statistics students may use a predefined set of notes. Please print the set of formula notes for the final. No other notes may be added to these pages. If any notes are added, the student will receive an F for cheating. Graphing Calculator
• Calculus Students: (Business Calculus, Calculus l - lll) - Calculus students may use a predefined set of notes. Please print the set of formula notes for the final. No other notes may be added to these pages. If any notes are added, the student will receive an F for cheating. -Graphing Calculator

The final exam must be proctored at college testing center or a Sylvan Learning Center. A valid driver's license or State ID must be shown at the testing center. An expired license or State ID will not be accepted.

The 60% rule was set in place to protect the integrity of online math education by requiring a display of competency in exchange for a grade. All schools which are regionally accredited adhere to online standards. Your college is accepting this course because it goes through a regionally accredited university, which tells your college that standards have been met. Your college will not accept a class from a school that is not regionally accredited, because they know the standards won't be met.

#### Assessment:

A 90-100 A Clearly stands out as excellent performance and, exhibits mastery of learning outcomes.
B 80-89 B Grasps subject matter at a level considered to be good to very good, and exhibits partial mastery of learning outcomes.
C 70-79 C Demonstrates a satisfactory comprehension of the subject matter, and exhibits sufficient understanding and skills to progress in continued sequential learning.
D 60-69 D Quality and quantity of work is below average and exhibits only partial understanding and skills to progress in continued sequential learning.
F 0-59 F Quality and quantity of work is below average and not sufficient to progress.

## Chapter 1

Basis Representation

Lessons Homework
1.1   The Principles of Mathematics Induction 1.1
1.2   The Basis Representation Theorem 1.2

## Chapter 2

The Fundamental Theorem of Arithmetic

Lessons Homework
2.1   Euclid's Division Algorithm 2.1
2.2   Divisibility 2.2
2.3   The Linear Diophantine Equation 2.3
2.4   The Fundamental Theorem of Arithmetic 2.4

## Chapter 3

Combinatorial and Computational Number Theory

Lessons Homework
3.1   Permuations and Combinations 3.1
3.2   Fermat's Little Theorem 3.2
3.3   Wilson's Theorem 3.3
3.4   Generating Functions 3.4
3.5   The Use of Computers in Number Theory 3.5

## Chapter 4

Fundamentals of Congruences

Lessons Homework
4.1   Basic Properties of Congruences 4.1
4.2   Residue Systems 4.2
4.3   Riffling 4.3

## Chapter 5

Solving Congruences

Lessons Homework
5.1   Linear Congruences 5.1
5.2   The Theorems of Fermat and Wilson Revisited 5.2
5.3   The Chinese Remainder Theorem 5.3
5.4   Polynomial Congruences 5.4

## Chapter 6

Arithmetic Functions

Lessons Homework
6.1   Combinatorial Study of phi(n) 6.1
6.2   Formulae for d(n) and sigma(n) 6.2
6.3   Multiplicative Arithmetic Functions 6.3
6.4   The Mobius Inversion Formula 6.4

## Chapter 7

Primitive Roots

Lessons Homework
7.1   Properties of the Residue System 7.1
7.2   Primitive Roots Modulo p 7.2

## Chapter 8

Prime Numbers

Lessons Homework
8.1   Elementary Properties of pi(x) 8.1
8.2   Tchebychev's Theorem 8.2
8.3   Some Unsolved Problems About Primes 8.3

## Chapter 9

Lessons Homework
9.1   Euler's Criterion 9.1
9.2   The Legendre Symbol 9.2
9.3   The Quadratic Reciprocity Law 9.3
9.4   Applications of the Quadratic Reciprocity Law 9.4

## Chapter 11

Sums of Squares

Lessons Homework
11.1   Sum of Two Squares 11.1
11.2   Sum of Four Squares 11.2

## Chapter 12

Elementary Partition Theory

Lessons Homework
12.1   Introduction 12.1
12.2   Graphical Representation 12.2
12.3   Euler's Partition Theorem 12.3
12.4   Searching for Partition Identities 12.4

## Chapter 13

Partition Generating Functions

Lessons Homework
13.1   Infinite Products as Generating Functions 13.1
13.2   Identities Between Infinite Series and Products 13.2
13.3   Euler's Partition Theorem 13.3

This course is online and your participation at home is imperative. A minimum of 8 - 10 hours per week of study time is required for covering all of the online material to achieve a passing grade. You must set up a regular study schedule. You have five months of access to your online account with a thirty-day extension at the end if needed. If you do not complete the course within this time line, you will need to enroll in a second term.

#### Schedule:

Below is the suggested time table to follow to stay on a 17 week schedule for the course. The following schedule is the minimum number of sections that need to be completed each week if you would like to finish in a regular semester time frame. You do not have to adhere to this schedule. You have five months of access plus a 30 day extension at the end if needed. You can finish the course as soon as you are able.

 Week Complete Sections 1 1.1 - 1.2 2 2.1 - 2.3 3 2.4 - 3.1 4 3.2 - 3.3 5 3.4 - 4.1 6 4.2 - 5.1 7 5.2 - 5.4 8 6.1 - 6.3 9 6.4 - 7.2 10 8.1 - 8.3 11 9.1 - 9.2 12 9.3 - 9.4 13 11.1 - 11.2 14 12.1 - 12.2 15 12.3 - 12.4 16 13.1- 13.2 17 13.3 Final Exam

## Conduct Code:

#### Code of Ethics:

Regulations and rules are necessary to implement for classroom as well as online course behavior. Students are expected to practice honesty, integrity and respect at all times. It is the student's responsibility and duty to become acquainted with all provisions of the code below and what constitutes misconduct. Cheating is forbidden of any form will result in an F in the class.

#### Respectful communications:

When contacting Omega Math or Westcott Courses, you agree to be considerate and respectful. Communications from a student which are considered by our staff to be rude, insulting, disrespectful, harassing, or bullying via telephone, email, or otherwise will be considered a disrespectful communication and will result in a formal warning.

We reserve the right to refuse service. If we receive multiple disrespectful communications from person(s) representing the student, or the student themselves, the student will be excluded from taking future courses at Westcott Courses/Omega Math.

#### Grading information and proctored final policies:

Cheating: Any form of cheating will result in an F in the class. If there is an associated college attached to the course, that college will be notified of the F due to cheating and they will determine any disciplinary action.

Any form of collaboration or use of unauthorized materials during a quiz or an exam is forbidden.

By signing up for a course, you are legally signing a contract that states that the person who is named taking this course is the actual individual doing the course work and all examinations. You also agree that for courses that require proctored testing, that your final will be taken at a college testing center, a Sylvan Learning center, and the individual signed up for this course will be the one taking the test. Failure to do so will be considered a breach of contract.

Other forms of cheating include receiving or providing un-permitted assistance on an exam or quiz; taking an exam for another student; using unauthorized materials during an exam; altering an exam and submitting it for re-grading; failing to stop working on the exam when the time is up; providing false excuses to postpone due dates; fabricating data or references, claiming that Westcott Courses/Omega Math lost your test and or quiz scores. This includes hiring someone to take the tests and quizzes for you.

#### Unauthorized collaboration:

Working with others on graded course work without specific permission of the instructor, including homework assignments, programs, quizzes and tests, is considered a form of cheating.

#### Important Notes:

This syllabus is subject to change and / or revision during the academic year. Students with documented learning disabilities should notify our office upon enrollment, as well as make sure we let the testing center know extended time is permitted. Valid documentation involves educational testing and a diagnosis from a college, licensed clinical psychologist or psychiatrist.

• Course Code: None

Transcript:

A certificate of completion is issued from Omega Math. This course under the non-credit option does not go through one of our partner universities; thus, a transcript is not included with the course.

Credits: 0

Certificate of Completion: Yes

Transfer:

If you would like to take this class for personal enrichment, the non-credit course is the exact same class as the credit course; it is just less expensive since it is not sent through our partner university for credit. If you want to transfer the course to your college, you will need to enroll under the semester credit option. If you would like pre-approval from your school, please send your counselor or registrar's office the link to this page. The non-credit courses can also be used to learn the material and then receive credit at a home college using Credit by Examination. (K-12 use)

Enrollment Schedule:

Enroll any day of the year, and start that same day. Students have five months of access, plus a 30 day extension at the end if needed. Students can finish the self-paced courses as soon as they are able. Most students finish the lower level courses in 4 - 8 weeks. The upper level math classes, such as Calculus and above, usually take students 3-4 months. (Note: The 30-day extension cannot take your total course time six months beyond the date of enrollment. At the end of the six months, we must post a grade with the university.)

Required Textbook:

Number Theory (Dover Books on Mathematics)

There is only one edition to this paperback, and it can be purchased on Amazon.
- George E. Andrews
- Textbook ISBN-10: 0-486-68252-8
- Textbook ISBN-13: 0-486-68252-5

Proctored Final: No

#### Description

Topics include: distribution of primes, representations of integers, Fibonacci numbers, divisibility, Euclidean algorithm, fundamental theorem of arithmetic, number-theoretic functions, Diophantine equations, congruence, primitive roots, the Chinese remainder theorem, quadratic residues, and elementary partition theory.
Prerequisite: Methods of Proof and Linear Algebra with a grade of C or better.

Note: This course is proof based. All homework assignments, exams and the final are graded by the instructor.

#### Learning Outcomes

At the conclusion of this course, students should be able to:

1. Apply divisibility properties and the Fundamental Theorem of Arithmetic.
2. Use the Euclidean algorithm to find the greatest common factor of two integers.
3. Find integral solutions to specified linear Diophantine equations.
4. Apply the Chinese remainder theorem.
5. Solve systems of linear congruencies.
6. Understand the basic properties of integers, including divisibility, primality, congruence, number theoretic functions, and their applications.
7. Apply Fermat's Little theorem to prove relations involving prime numbers.
8. Use Wilson's theorem to solve congruencies involving factorials.
9. Apply the Mobius Inversion formula.
10. Understand current unsolved problems about primes and their importance to the field of mathematics and other related fields.

#### Methods of Evaluation:

Homework 15%
Exams 60%
Final 25%
(You must get at least 60% on this final in order to pass the class with a C or better.)

Homework:
Homework assignments are essential in a mathematics course. It is not possible to master the course without a considerable amount of time being devoted to studying the concepts and using the concepts to solve problems. Each lesson contains a set of homework problems.

Exams:
The exams are designed to cover a broader area of the text and test your understanding of the material. Typically, exams will cover two chapters of the text.

#### Assessment:

A 90-100 A Clearly stands out as excellent performance and, exhibits mastery of learning outcomes.
B 80-89 B Grasps subject matter at a level considered to be good to very good, and exhibits partial mastery of learning outcomes.
C 70-79 C Demonstrates a satisfactory comprehension of the subject matter, and exhibits sufficient understanding and skills to progress in continued sequential learning.
D 60-69 D Quality and quantity of work is below average and exhibits only partial understanding and skills to progress in continued sequential learning.
F 0-59 F Quality and quantity of work is below average and not sufficient to progress.

## Chapter 1

Basis Representation

Lessons Homework
1.1   The Principles of Mathematics Induction 1.1
1.2   The Basis Representation Theorem 1.2

## Chapter 2

The Fundamental Theorem of Arithmetic

Lessons Homework
2.1   Euclid's Division Algorithm 2.1
2.2   Divisibility 2.2
2.3   The Linear Diophantine Equation 2.3
2.4   The Fundamental Theorem of Arithmetic 2.4

## Chapter 3

Combinatorial and Computational Number Theory

Lessons Homework
3.1   Permuations and Combinations 3.1
3.2   Fermat's Little Theorem 3.2
3.3   Wilson's Theorem 3.3
3.4   Generating Functions 3.4
3.5   The Use of Computers in Number Theory 3.5

## Chapter 4

Fundamentals of Congruences

Lessons Homework
4.1   Basic Properties of Congruences 4.1
4.2   Residue Systems 4.2
4.3   Riffling 4.3

## Chapter 5

Solving Congruences

Lessons Homework
5.1   Linear Congruences 5.1
5.2   The Theorems of Fermat and Wilson Revisited 5.2
5.3   The Chinese Remainder Theorem 5.3
5.4   Polynomial Congruences 5.4

## Chapter 6

Arithmetic Functions

Lessons Homework
6.1   Combinatorial Study of phi(n) 6.1
6.2   Formulae for d(n) and sigma(n) 6.2
6.3   Multiplicative Arithmetic Functions 6.3
6.4   The Mobius Inversion Formula 6.4

## Chapter 7

Primitive Roots

Lessons Homework
7.1   Properties of the Residue System 7.1
7.2   Primitive Roots Modulo p 7.2

## Chapter 8

Prime Numbers

Lessons Homework
8.1   Elementary Properties of pi(x) 8.1
8.2   Tchebychev's Theorem 8.2
8.3   Some Unsolved Problems About Primes 8.3

## Chapter 9

Lessons Homework
9.1   Euler's Criterion 9.1
9.2   The Legendre Symbol 9.2
9.3   The Quadratic Reciprocity Law 9.3
9.4   Applications of the Quadratic Reciprocity Law 9.4

## Chapter 11

Sums of Squares

Lessons Homework
11.1   Sum of Two Squares 11.1
11.2   Sum of Four Squares 11.2

## Chapter 12

Elementary Partition Theory

Lessons Homework
12.1   Introduction 12.1
12.2   Graphical Representation 12.2
12.3   Euler's Partition Theorem 12.3
12.4   Searching for Partition Identities 12.4

## Chapter 13

Partition Generating Functions

Lessons Homework
13.1   Infinite Products as Generating Functions 13.1
13.2   Identities Between Infinite Series and Products 13.2
13.3   Euler's Partition Theorem 13.3

This course is online and your participation at home is imperative. A minimum of 8 - 10 hours per week of study time is required for covering all of the online material to achieve a passing grade. You must set up a regular study schedule. You have five months of access to your online account with a thirty-day extension at the end if needed. If you do not complete the course within this time line, you will need to enroll in a second term.

#### Schedule:

Below is the suggested time table to follow to stay on a 17 week schedule for the course. The following schedule is the minimum number of sections that need to be completed each week if you would like to finish in a regular semester time frame. You do not have to adhere to this schedule. You have five months of access plus a 30 day extension at the end if needed. You can finish the course as soon as you are able.

 Week Complete Sections 1 1.1 - 1.2 2 2.1 - 2.3 3 2.4 - 3.1 4 3.2 - 3.3 5 3.4 - 4.1 6 4.2 - 5.1 7 5.2 - 5.4 8 6.1 - 6.3 9 6.4 - 7.2 10 8.1 - 8.3 11 9.1 - 9.2 12 9.3 - 9.4 13 11.1 - 11.2 14 12.1 - 12.2 15 12.3 - 12.4 16 13.1- 13.2 17 13.3 Final Exam

## Conduct Code:

#### Code of Ethics:

Regulations and rules are necessary to implement for classroom as well as online course behavior. Students are expected to practice honesty, integrity and respect at all times. It is the student's responsibility and duty to become acquainted with all provisions of the code below and what constitutes misconduct. Cheating is forbidden of any form will result in an F in the class.

#### Respectful communications:

When contacting Omega Math or Westcott Courses, you agree to be considerate and respectful. Communications from a student which are considered by our staff to be rude, insulting, disrespectful, harassing, or bullying via telephone, email, or otherwise will be considered a disrespectful communication and will result in a formal warning.

We reserve the right to refuse service. If we receive multiple disrespectful communications from person(s) representing the student, or the student themselves, the student will be excluded from taking future courses at Westcott Courses/Omega Math.

#### Grading information and proctored final policies:

Cheating: Any form of cheating will result in an F in the class. If there is an associated college attached to the course, that college will be notified of the F due to cheating and they will determine any disciplinary action.

Any form of collaboration or use of unauthorized materials during a quiz or an exam is forbidden.

By signing up for a course, you are legally signing a contract that states that the person who is named taking this course is the actual individual doing the course work and all examinations. You also agree that for courses that require proctored testing, that your final will be taken at a college testing center, a Sylvan Learning center, and the individual signed up for this course will be the one taking the test. Failure to do so will be considered a breach of contract.

Other forms of cheating include receiving or providing un-permitted assistance on an exam or quiz; taking an exam for another student; using unauthorized materials during an exam; altering an exam and submitting it for re-grading; failing to stop working on the exam when the time is up; providing false excuses to postpone due dates; fabricating data or references, claiming that Westcott Courses/Omega Math lost your test and or quiz scores. This includes hiring someone to take the tests and quizzes for you.

#### Unauthorized collaboration:

Working with others on graded course work without specific permission of the instructor, including homework assignments, programs, quizzes and tests, is considered a form of cheating.

#### Important Notes:

This syllabus is subject to change and / or revision during the academic year. Students with documented learning disabilities should notify our office upon enrollment, as well as make sure we let the testing center know extended time is permitted. Valid documentation involves educational testing and a diagnosis from a college, licensed clinical psychologist or psychiatrist.