Welcome to Math 261, Introduction to Number Theory.

**Instructor: ** Cornelia Yuen

**Texts:** *Elementary Introduction to Number Theory*, Third Edition,
by Calvin T. Long.

**Course Content:** We will study the basic concepts, ideas, results,
and algorithms of number theory, covered in Chapters 1-5 of the text (but not
in that order). We will also cover some supplementary topics. Students should
know the precise definitions and statements of theorems and concepts.

**Group work**: Although I will do a significant amount of lecturing,
we will also work at times in groups of three or four in class on
exploratory problems, discovery exercises, and skill development. You will
also occasionally do in-class presentations with the support of your
groupmates.

**Graded homework:** Homework will be assigned regularly. The
problems sets must be handed in class on the due day of the work.

**Quizzes:** Quizzes will be given, usually with advance warning,
every two weeks or so, but not on a rigidly regular schedule.

**Exams:** We will have two midterms and one cumulative final
exam, spaced roughly equally throughout the semester. Of course, it is
a very bad idea to be absent on an exam day. But in some cases,
especially serious illness and prior arrangement, excused absences can
be warranted.

**Grade weighting:** You will be graded in three areas:
homework (30%), quizzes (15%), and exams (15%,20%,20%). Class participation
and attendance will be graded only indirectly, by means of exams.

**Attendance:** Class attendance is mandatory.

Number theory is the branch of mathematics which studies the properties of numbers. Carl Friedrick Gauss (1777-1855), often known as the "prince of mathematics", once remarked that mathematics is the "queen of sciences" but number theory is the "queen of mathematics".

There are several fields of number theory. In this course, we study elementary number theory, in which elementary methods (e.g. arithmetic, high school algebra) are used to solve equations with integeral or rational solutions. Some of the major topics/techniques we will explore include

- mathematical induction
- greatest common divisors and the Euclidean algorithm
- Fundamental Theorem of Arithmetic
- primes and unique factorization
- congruence relation
- Chinese remainder theorem
- law of quadratic reciprocity

Many questions in number theory can be stated in elementary number
theoretic terms, but their answers may not be as easy as the problems
first suggest, and they may fall outside the realm of elementary number
theory. One of the most famous examples is Fermat's last theorem (there is
no integral solution for the equation
x^{n}+y^{n}=z^{n} for n > 2), which
was stated in 1637 but only recently proved in 1994. Another example is
the Goldbach's conjecture (every even number can be written as the sum of
two primes), which was proposed in 1742 and remains unproven.

**HW#1**(due Wed 1/24)**HW#2**(due Fri 2/2)**HW#3**(due Fri 2/16)**HW#4**(due Mon 2/26)**HW#5**(due Fri 3/9)**HW#6**(due Fri 3/30)**HW#7**(due Wed 4/11)**HW#8**(optional)

One important way to succeed is to know the precise definitions and statements of theorems and concepts. Please make an attempt to digest and retain these at all times.

**Quiz #1**(Mon 1/29)- Summation and product, mathematical induction, and well-ordering principle (roughly Sections 1.1, 1.4, 1.5 in the textbook).
- Click here for the solution.

**Exam #1**(Fri 2/9)- Lectures from the beginning through 2/5 -- Summation and product, mathematical induction, well-ordering principle, division algorithm, greatest common divisor, and Eucldiean algorithm.
- If you are interested, click here for the review sheet.
- Extra office hours for Exam #1:
- M 11am-12:30pm, 1-2pm, 3-4pm
- T 10:30am-noon
- W 11am-noon, 1-2pm, 3-3:30pm

We will have a review session in class on Wednesday 2/7. This is not another hour for me to lecture, but rather an opportunity for you to ask questions which may have come up while you are studying for the exam. If you have any questions/concern/difficulty regarding specific homework problems, exercises in the textbook and/or material we covered in class, you are most welcome to bring them with you.

**Quiz #2**(Wed 2/21)- Linear diophantine equations and least common multiples.
- For more exercises on linear diophantine equations, try #2, 3, 5 in Exercises 5.1 in the textbook (click here if you don't have the textbook).
- Least common multiples is covered in Section 2.4 in the textbook.
- Click here for the solution.

**Quiz #3**(Mon 3/5)- Prime numbers, Mersenne, Fermat and Perfect numbers (roughly sections 3.1, 3.2 (through Theorem 3.3), 3.4 in the textbook).
- Click here for the solution.

**Exam #2**(Fri 3/23)- Lectures after Exam #1 through 3/21 -- Linear diophantine equations, least common multiples, prime numbers, Mersenne, Fermat and perfect numbers, equivalence relations, basics of congruence of integers, and divisibility tests.
- If you are interested, click here for the review sheet.
- Extra office hours for Exam #2:
- M 11am-12:30pm, 1-2pm, 3-4pm
- T 10:30am-noon
- W 11am-noon, 1-2pm, 3-3:30pm

**Quiz #4**(Wed 4/4)- Finding day of the week, congruence cancellation/divison and Fermat's Little Theorem (Theorem 4.6 through Corollary 4.10 in Section 4.1, and Theorem 4.16 and Corollary 4.18 in Section 4.3 in the textbook).
- Click here for the solution.

**Quiz #5**(Wed 4/18)- Euler phi-function, reduced residue systems, Euler's Theorem and linear congruences (Sections 4.3 and 5.1 in the textbook).
- Click here for the solution.

**Final Exam**(Wed 5/2, 3:30-5:30pm)- This will be an accumulative test, in other words, it will cover all the material discussed throughout the semester. However, material covered after the second exam will weigh slightly more.
- If you are interested, click here for the review sheet.
- Extra office hours during last week of class (4/23-27):
- M 11am-noon, 1-2pm, 3-4pm
- W 11am-noon, 1-2pm, 3-4pm
- Th 10:30am-noon
- F 11am-noon, 1-2pm, 3-4pm

(The prime numbers, arranged in a spiral)