Number theory starts off with the study of whole numbers, the most basic objects in mathematics. So number theory is one of the oldest branches of mathematics. Ancient Greeks, Egyptian, and Babylonians studied basic problems about numbers, such as whether there are infinitely many prime numbers, or how to find integer solutions of equations like x2 + y2=5z2. See Saint Andrews University page on Pythagoras's theorem in Babylonian mathematics for pictures of some of the oldest known number theory.
Because whole numbers are discrete, i.e, not continuous, e.g., there is no whole number between 0 and 1, number theory is very relevant for computers, which work digitally, i.e., also with discrete quantities, e.g., 0 and 1.
Because of this, Number theory is the basis of important computer applications, such as cryptography, used to transfer data securely over the internet, and coding theory, used to transmit data, e.g., from your CD to your CD player, in such a way that errors in transmission can be corrected.
So, as well as being an old area of mathematics, number theory is a very important and active area of modern mathematical research.
Number theory is a very broad area. This course will concentrate on algebraic aspects of number theory. There will be some overlap with some topics covered in the abstract algebra course, MATH 4200, but nothing about abstract algebra will be assumed, though if you are taking that course, you may find this course gives useful extra examples of topics covered in MATH4200.
In this course, we start with studying the integers, and then move on to studing numbers modulo n, denoted Zn. These form the basis of RSA cryptography, and also have many other applications in mathematics. The study of Zn will take up most of the course (this is chapters II, III, IV of the text book). We should then have time to study one further topic, sums of squares, chapter V.
Approximate schedule:
We will closely follow the text book, in the order it's written.
Please try and read and
read ahead for each class. This should mean reading on average about two or three pages before
each class (there are about 40 classes, and we cover about 80 pages of the text book).
Please try and read every word, and make a note of what you don't understand,
so you can ask about it in class.
Expect to cover between 1 and 2 sections per week, depending on
the lengths of the sections.
A strong student who is intending to study number theory at
graduate level should aim to work through all the excersise in each section, though probably
we will not have enough time to go over every exercise in class.
There are many many web sites on number theory, reading around will give you different
perspectives, so if you have time this is also a good idea.
This is a pure mathematics course, and proofs will be given, and students will be expected
to be able to learn to read and
understand proofs and reproduce proofs in tests and exams.
If there is time, application to RSA cryptography will be explained, but this will not
be examined.