Number Theory (FA22)

Number 1

Number 2

Number 3

Number 4

Midterm 1 solutions:

Midterm solutions

Midterm Redux solutions

Practice Midterm 1:

Practice Midterm (Update: Question 3 had a typo)

Practice Midterm Solutions

Solutions to Midterms/Assignments:

Midterm 0 (with solutions)

Solutions to Assignment 1

Solutions to Assignment 3

Solutions to Assignment 5

Course Catalogue Description:

The great mathematician C. F. Gauss once wrote "Mathematics is the queen of sciences and number theory is the queen of mathematics." Number theory is that part of mathematics dealing with the integers and certain natural generalizations. Topics include modular arithmetic, unique factorization into primes, linear Diophantine equations, and Fermat's Little Theorem. Discretionary topics may include cryptography, primality testing, partition functions, multiplicative functions, the law of quadratic reciprocity, historically interesting problems.

Learning Outcomes:

By the end of this course, you should be able to:

1. Understand of the basic structures of number theory: define terms, explain their significance, and apply them in context;
2. Solve mathematical problems: utilize abstraction and think creatively;
3. Learn the basics of mathematical communication and logical arguments. Write clear mathematical proofs, recognize and construct mathematically rigorous arguments; and
4. Learn about the role of computation within modern mathematics.

Course Expectations

1. Proofs and proof techniques

Using mathematics at a higher level is far more than "solving for x", "knowing formula Y", or "using recipe Z to solve a word problem". Really, mathematics is more about ideas and how they connect together. As the ideas become more sophisticated, so too evolves the language and conventions needed to express these ideas.

Math 25 is one of the first courses that really get into the internal structure of "how math works". Proofs and mathematical reasoning are at the core of mathematics, and this course largely serves as an introduction to those themes. Let's consider the following statements:

• 40 is not the sum of three cubes
• 41 is not the sum of three cubes
• 42 is not the sum of three cubes

One could test these statements by trying values up to 1,000,000 for each variable and reasonably expect that all three results are true. However, one of them actually is the sum of three cubes! So long as one accepts a small number of properties of the integers that "surely must be true" (formally, axioms), math gives us a way to decide which of these statements are actually true, and which are not. (Let us for simplicity ignore the incompleteness theorems.)

2. Core ideas of elementary number theory.

In addition to proofs, this course is an introduction to the wonderful world of number theory. We explore the topics of the natural numbers from an "elementary'' standpoint. By "elementary", we mean what can be said about the integers, and the prime numbers, without appealing to the more modern areas of math like complex analysis, algebraic geometry, or group theory. (Disclosure: there will be a tiny bit of group theory.)

As the ancients did we begin with the integers. Multiplication and division lead us to the notions of divisibility, primes, and of a greatest common divisor. Asking about remainders and their structure leads to congruences. Congruences lead to the Chinese Remainder Theorem. So on it goes.

We will very briefly mention some of the connections between number theory and cryptography, which is a cornerstone technology for the modern internet. I suppose G.H. Hardy might have been appalled at the sullying of "pure mathematics", but the memes were worth it.

At the end of the course we'll explore some extension of the ideas discussed during the term.

3. Computation in mathematics.

Computation is increasingly more important within scientific disciplines. This is certainly true within mathematics as well, and new ideas are discovered every day with the use of computers that would be hopelessly inaccessible without their help. Thus, one of the aims of this course is to introduce how computers are used to explore mathematical questions.

All of this said, the point of the course is not to learn computer programming. All mandatory programming tasks will be done in groups.

Note: If you wish to use an algorithmic approach on an individual assignment, please ask your instructor first. Sometimes, the point of a question is to refine your knowledge of a particular concept (example: compute gcd(123,432) using the Euclidian algorithm) -- the python call ``gcd(123, 432)`` will compute the correct answer, but not further your understanding.

Course assessment structure:

In this course, there will be

• 3 Group programming assignments (15%)
• 7 Individual written assignments (30%)
Note: Individual assignments on weeks concurrent with a group assignment are shorter.
• 1 Midterm (20%)
• 1 Final exam (35%)