Welcome to Number Theory!
Number theory is a branch of mathematics which stretches from the simplest facts about whole numbers to investigations which are as deep and as difficult and as sophisticated in their approach as any part of Mathematics.
This course will restrict itself to fairly straightforward investigations on properties of numbers which can be dealt with by 'elementary means'. Thus we will not use more advanced mathematical techniques such as calculus or abstract algebra. However, 'elementary' does not always mean 'easy' and these elementary techniques will often reach deep into the nature of natural numbers.
Many of the topics we discuss have found important recent applications in cryptography and we shall discuss these from time to time.
There are course notes which will be available
for sale from the Bookroom. The Notes
are also available free in pdf format on the web.
If you feel a need for additional reading on the topic then I
particularly recommend
Number theory with computer applications
by
Ramanujachary Kumandari and Cristina Romero
Prentice Hall (1998)
which is well-written and has an excellent account of the recent computer applications.
There are many other books on the topic. You should look for titles of the general form '..Introduction to Number Theory...' or '..Elementary Number Theory...'
So how do you do large scale computation in number theory? You need, as well as a computer or calculator, a programming environment that will enable you to calculate with arbitrary precision (so that, for example, 2 to the power of 100 comes out as an integer and not a floating point decimal). There are several ways to get this.
One way is to use one of the symbolic computation programs Mathematica or Maple. If you own one of these then there is no need to look further. But they are rather expensive, even in the cheap student editions.
Another way, which I strongly recommend, is to use a free program, Pari-Gp. It is very fast on number-theoretic problems and has a programming environment that is designed for number theory. It also seems fairly reliable and has comprehensive documentation.
Pari-Gp is available on machines in both of the computer labs. The computer lab (G70 or the Wilson Lab.) is also booked at all practice class times (that is on Mondays at 3.15). On the second week of the semester at this time, we will have an informal tutorial on the use of Pari-Gp.
If you wish to save programs in text files, then you can save them in the folder "GP-Data" within the Applications folder (say as "prog.txt"). Then you can read them in to GP using the command "\r prog.txt" (no quotes). I will give more details when we have the practice class in the lab.
I have downloaded the manual which you can look at using a pdf viewer. I strongly suggest, however, that you first read the section of the Notes devoted to Pari-Gp. The program contains many more functions than we can use and the manual can be a little daunting if you approach it with no prior knowledge.
You can also obtain a free copy of Pari-Gp to use on your own machine; it is available for Mac (System 9), for PC and (in its most natural form) for Unix based machines. For Mac OSX, Max Flander has donated this set of simple instructions.
History
of Mathematics archive
A very good (and award winning) collection of material on the history
of mathematics. Highly recommended.
Mersenne Prime Search
The home page for a collaborative internet search for new Mersenne primes
The Prime Puzzles & Problems Connection
A collection of puzzles and problems connected with primes.
The Prime
Page (prime number lists, records, news, definitions...)
A huge amount of information about primes. Includes, for example, lists
of primes and software related to primes.
The
Prime Machine
An on-line program for calculating and counting primes.
General article on Public Key Codes
An article at an elementary level.
General article on primes
An article at an elementary level.
The Clay Millenium problems
These problems have been compiled by a board of eminent mathematicians and there is a prize of US$1,000,000 for the solution of each one. The Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture have special significance for Number Theory.
Mathematics
Archives - Topics in Mathematics - Number Theory
A large collection of links to number theory sites. Some of the other sites
are listed here
If you don't like something about the course, tell me. If you do, also tell me. It all helps to improve things. I am in Room 171 in the Department of Mathematics and Statistics.
The course last year was generally well-received. A few people have some problems with the practice class but I hope that these can be solved with some more encouragement to attend and participate.
My office hours for 351 are Wednesday at 2.15; but you are welcome to try to find me at other times.
I can also be contacted by email. (But if you have mathematical questions it is generally easier to deal with them by coming to see me.)
The SSLC representative for the course is Aaron Lau.
Assignment 1 (out Wednesday 20 August)
Midsemester test (Wednesday 10 September)
Assignment 2 (out Wednesday 15 October)
To view any of these, you will need the Adobe Acrobat
Reader. If there is not one already on the machine you are using, it
can be obtained free from Adobe.
It is simple and convenient to use.
Notes for 620-351:Number Theory
This is a list of Pari_GP programs which implement the algorithms in the notes. They have not been carefully tested. Use them at your peril.
Misprints; please email me if you notice any misprints.
No too much to say really.
A few people commented that they did not find Pari-Gp useful. I believe it can be useful. If you have any programming skills then I believe that trying a few of the programming exercies will greatly enhance your understanding. If you don't have any programming skills then it can at least give you a handy "calculator" for checking examples. Some understanding of the computational aspects of what we do is essential to understanding applications.
The computational problems also give an alternative way through the subject for those who like to avoid proofs.
In addition to learning specific technical skills that will
assist you in your future careers in science, engineering, commerce, education
or elsewhere, you will have the opportunity to develop in this subject, generic
skills that will assist you whatever your future career path.