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RECREATIONAL NUMBER THEORY
This web-site is intended to describe several examples of the use of elliptic curves to attack problems from the area of recreational number theory - see the following link for recent news.
These two areas are sometimes thought of as being at opposing poles in Mathematics. Elliptic curves can provide some of the deepest results, such as Andrew Wiles' proof of Fermat's Last Theorem, whilst recreational number theory is enjoyed by many without any formal Maths qualifications.
It is hoped that the following pages can show that there can be a fruitful mixing between the two topics. Most of the problems and their analysis are not original to me. I have tried to give the correct credit - if I have missed you out, my sincere apologies.
The choice of examples is purely subjective with no attempt to claim that it is exhaustive. The only criterion for inclusion is that I found the problem interesting enough to play around with it. If any reader can suggest an interesting additional example, please send me an e-mail.
Currently, the examples fall into four distinct categories:
The basic methodology involved in attacking these
problems is described in this page .
I also have a file of references
to books and papers, related to the problems considered and other topics,
which might be of interest to the reader.
WARNING: The following pages all have a lot of
symbols. There are a large number of web-browsers currently available,
and I cannot check that the pages look OK on all of them. I have tried
the pages on Internet Explorer, Netscape, and Mozilla Firefox, though there
could still be errors.
N = ( x + y + z )*( 1/x + 1/y + 1/z ) |
N = | ( x + y + z )^{3 }x y z |
N = | x
y |
+ | y
z |
+ | z
x |
N = | ( x^{ }+ y + z )^{3 } x^{3 }+ y^{3} + z^{3} |
These are elliptic curves given directly
with no seeming relation to an underlying problem.
Computing Resources:
As will be seen from the results quoted in the various
pages, this type of Maths needs a computer to get anywhere beyond first-base.
A description of what I use can be found here
.
Contact information:
If you especially like or dislike anything in these pages you can contact me as follows
e-mail: allan.macleod@paisley.ac.uk
postal mail: Allan MacLeod
Dept. of Mathematics and Statistics
University of Paisley
High St.
Paisley
SCOTLAND
PA1 2BE
telephone: (within UK) 0141-848-3516
(outwith UK) 44-141-848-3516
Disclaimer (to keep my employers happy!)
Since much of these pages isn't original to me, anybody
can use the material in any way they want, without needing to seek my permission.
I will be delighted if someone uses the information. I have tried to ensure
there are as few mistakes as possible, but it is inevitable that some errors
will have eluded me. If you find one, send me an e-mail please. There is,
thus, no guarantee that the data is correct, and you use it at your own
peril.