“More Fun with Prime Numbers”
This is the second round of the course as the self-paced format. The original version of the course was produced and operated from November 2,
Instructor: Tetsushi Ito
See "Meet the Course Staff" section for more details.
Course Description
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 are all prime numbers and they hold special significance. Mathematicians from ancient times to the 21st century have been working on prime numbers, as they're one of the most mysterious and important subjects in mathematics.
In this course, I will present several attractive topics on prime numbers. You will learn basic concepts of prime numbers from the beginning. They obey mysterious laws. Some laws are easily verified by hand, some laws were discovered 100 years ago, and some laws are yet to be discovered. Surprisingly, prime numbers are also applied to cryptography today. You will also learn how to construct practical cryptosystems using prime numbers.
The original course "Fun with Prime Numbers" was first offered in 2015 and attracted many students. This course in 2017 will be offered as its refined and upgraded version. All the lecture videos have been renewed, and a new topic on cryptography has been added so as to enliven and satisfy even the students who took the previous course.
No previous knowledge of prime numbers is required in this course. Calculating with a pen and paper, you will explore the mysterious world of prime numbers. The course is designed to encourage you to attack unsolved problems, and hopefully, discover new laws of your own in the future!
What you'll learn
- Basic Properties of Prime Numbers
- Modular Arithmetic and Fermat's Little Theorem
- Laws of Prime Numbers
- Applications of Prime Numbers to Cryptography
- Open Problems and Recent Advances
Prerequisites
Secondary school (high school) level algebras; basic mathematics concepts
Time commitment
2-3 hours per week
Lectures
Each course will be provided with short Lecture Videos by the instructor Tetsushi Ito along with a set of Problems related to the contents of the Lecture Videos. By watching the videos and answering the Problems, we hope that all participants will gain some basic knowledge
To get started, click on the "Course" tab at the top of the page.
Discussion forum
You are invited to participate in the Discussion forum (See Forum Guidelines here) to share ideas and ask questions to peers relating to each of the course’s contents. We hope this opportunity will lead to fruitful exchanges and discussion.
Assignments and Grading Criteria
To earn a certificate for the course, students must mark the score of 60% or more. Grading for the course is as below.
A: 85 -100%
B: 75 - 84%
C: 60 - 74%
F: Below 59%
If you are on the verified track and mark the passing score, certificates will be issued automatically by
Problems assigned every week, count for 40% (8% for Problems of each week) in total, respectively. During this course, learners are asked to work on five Homework assignments. The total of Homework counts for 40% (8% for Homework of each week). In Week 5, learners are encouraged to take the Final Exam (20%).
Problems: 40%
- Due date: End of course
Homework: 40%
- Due date: End of course
Final Exam: 20%
- Due date: End of course
Please pay attention to due dates of each Problem, Homework, and so on. To avoid any kinds of unexpected troubles including the Internet disconnection, we strongly recommend all learners to submit them with time to spare.
Course Schedule
|
Week |
Topic |
Homework |
|
1 |
What are Prime Numbers? Introduction to basic concepts and properties of prime numbers, such as infinitude of prime numbers, counting prime numbers, and the Basel problem and its relation |
Yes |
|
2 |
Sums of Two Squares Introduction to the modular arithmetic and its applications to number theory, including Fermat's Little Theorem, Wilson's Theorem, and Fermat's Theorem on Sums of Two Squares. |
Yes |
|
3 |
The Reciprocity Laws Introduction to the Quadratic Reciprocity Laws proved by Gauss. Several generalizations of the Quadratic Reciprocity Laws are also explained. |
Yes |
|
4 |
Prime Numbers and Cryptography Introduction to cryptography, and the construction practical cryptosystems using prime numbers. More recent topics on elliptic curve cryptosystems are also explained. |
Yes |
|
5 |
Introduction to several open problems and conjectures on prime numbers, including the Birch and Swinnerton-Dyer conjecture and the ABC conjecture. |
Yes |
This course will end on Thursday, March 14, 2019.