This course will explore the beauty and mystery of mathematics through a study of the patterns and properties of the natural numbers 1,2,3... . We'll discuss various special classes of numbers, like Fibonacci numbers, factorials and binomials and the many ways they arise in mathematics and nature. We'll also investigate the mysterious behavior of the prime numbers and their distribution, and alternative counting systems such as modular arithmetic.
Prerequisites
We will assume no mathematical background beyond high school algebra. Emphasis will be placed on discovery through conjecture and experimentation.
Faculty
| Name | Office | E-mail address |
| Noam Elkies | Science Center 335 | elkies@math.harvard.edu |
| Barry Mazur | Science Center 512 | mazur@math.harvard.edu |
Teaching Fellows
| Name | Office | E-mail address |
| Grigor Grigorov | Science Center 431h | grigorov@math.harvard.edu |
| Florian Herzig | Science Center 431e | herzig@math.harvard.edu |
| Robert Neel | Science Center 428d | rwneel@math.harvard.edu |
Course Website
www.courses.fas.harvard.edu/~qr28
Course Texts
Benedict Gross and Joseph Harris, The Magic of Numbers.
The course text is available for purchase at the Coop.
Homework
There will be short homework assignments after each lecture. The homework will be designed to review concepts from the previous lecture, to reinforce topics in the reading and to introduce some of the ideas of the next lecture. Grades for the homework will be as follows: check-plus for outstanding work; check for good work; check-minus for satisfactory work; minus for unsatisfactory or no work. All of the homework questions will be posted on the course's website; they will not be handed out in class. Homework is due the lecture after it is assigned. Absolutely no late homework will be accepted, but the lowest three homework grades will be discarded at the end of the semester.
Exams
There will be two midterm examinations during the semester and a final
exam during finals period. The midterms will be given during class
on Wednesday, March 2nd and Wednesday, April 13th.
If you have a conflict with either of the exam times, notify one of the TF's
immediately. The final exam is scheduled for Wednesday, May 25 and is administered by the registrar's office.
Grading Policy
The final grade will be based on the homework, the two midterms, and the final. The homework and each midterm will count for 20% of the final grade, and the final exam will count for 40% of the final grade. Minor adjustments may be made to take into account improvement during the semester.
Sections
Section times will be announced in the second week of the course, based upon the students' schedules. These will meet for one hour per week. In section we will review some topics from lecture and explore related issues. Section attendance is required. We will ask you on Monday, February 7th, for the openings in your schedules and we will e-mail your section assignments to you once they are available. Sections will begin meeting during the second week of classes.
Schedule
Below is the tentative lecture schedule for the course. It may change somewhat depending on the length of time required for certain topics and student interest.
Wednesday, February 2: The Remarkable Fibonacci Numbers
Friday, February 4: How to Count Without Counting I
Gross-Harris, Chapter 1
Monday, February 7: How to Count Without Counting II
Gross-Harris, Chapter 2
Wednesday, February 9: How to Count Without Counting III
Gross-Harris, Chapter 3
Friday, February 11: Counting Collections of Objects
Gross-Harris, Chapter 4
Monday, February 14: Probability
Gross-Harris, Chapter 5
Wednesday, February 16: Pascal's Triangle
Gross-Harris, Chapter 6
Friday, February 18: Finish and Recap Counting and Probability
Monday, February 21: President's Day (no class)
Wednesday, February 23: The Euclidean Algorithm
Gross-Harris, Chapter 8
Friday, February 25: The Euclidean Algorithm II
Gross-Harris, Chapter 8
Monday, February 28: Review
Wednesday, March 2: First Midterm Exam
Friday, March 4: Diophantine Equations I
Gross-Harris, Sections 9.1-9.2
Monday, March 7: Diophantine Equations II
Gross-Harris, Sections 9.3-9.4
Wednesday, March 9: Prime Numbers
Gross-Harris, Chapter 10
Friday, March 11: Unique Factorization
Gross-Harris, Chapter 11
Monday, March 14: Consequences of Unique Factorization
Gross-Harris, Chapter 12 and Section 13.1
Wednesday, March 16: Numbers and Number Systems
Gross-Harris, Chapter 14
Friday, March 18: Modular Arithmetic I
Gross-Harris, Chapter 15
Monday, March 21: Modular Arithmetic II
Gross-Harris, Chapter 16
Wednesday, March 23: Modular Arithmetic III
Gross-Harris, Chapter 17
Friday, March 25: Fermat's Little Theorem
Gross-Harris, Sections 18.1-18.5
March 26 through April 3: Spring Break (no class)
Monday, April 4: Computing Powers Modulo p
Gross-Harris, Sections 18.6-18.7
Wednesday, April 6: Computing Roots Modulo p, part I
Gross-Harris, Chapter 19
Friday, April 8: Computing Roots Modulo p, part II
Gross-Harris, Chapter 19
Monday, April 11: Review
Wednesday, April 13: Second Midterm Exam
Friday, April 15: Euler's Theorem
Gross-Harris, Sections 13.2-13.4 and Section 20.1
Monday, April 18: Computing Powers Modulo m
Gross-Harris, Sections 20.2-20.4
Wednesday, April 20: Computing Roots Modulo m I
Gross-Harris, Section 20.4
Friday, April 22: Computing Roots Modulo m II
Gross-Harris, Section 20.4
Monday, April 25: How to Build Codes I
Gross-Harris, Chapter 21
Wednesday, April 27: How to Build Codes II
Gross-Harris, Chapter 22
Friday, April 29: Distribution of Primes
Gross-Harris, Sections 23.1-23.2
Monday, May 2: Primality Tests I
Gross-Harris, Sections 23.3-23.6
Wednesday, May 4: Primality Tests II
Gross-Harris, Sections 23.7-23.9
Friday, May 6: Generators Modulo p
Gross-Harris, Chapter 24