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:

1. Representations of integers as specified combinations of integers.

 
2. Geometric properties of triangles having an integer value.

 
3. Problems inspired by reading volume 2 of Dickson's "History of the Theory of Numbers". This is the volume on Diophantine Analysis.

 
4. Classic problems where elliptic curves can help one find parametric solutions or partial solutions.

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.
 
 

1  Integer Representation Examples

1. Twocubes Problem . Find two rational numbers X and Y such that
   
                                             X³ + Y³ = N
   
   
   
2. Knight's Problem . Find integers x,y,z such that



 

N = ( x + y + z )*( 1/x + 1/y + 1/z )


 
3. Find integers x,y,z such that



 

N =

( x + y + z )3  x y z


 
4. Find integers x,y,z such that



 

N =

x  y

+

y  z

+

z  x


 
5. Recently, I have been investigating the form



 

N =

( x + y + z )3   x3 + y3 + z3

2  Geometric Properties

1. Find a rational right-angle triangle with area N. This is equivalent to the famous Congruent Number problem.

 
2. Find a rational triangle with one angle = 2 ? / 3 or ? / 3 and area = N ? 3, for an integer N which we can assume to be squarefree.

 
3. Find two integer right-angle triangles with the same base, and altitudes in the ratio N.

 
4. Find an integer triangle such that base/altitude or altitude/base equals N.

 
5. Find an integer triangle, with square integer area, and tan(C/2)=N or 1/N .

3  Dippings from Dickson

1. The Concordant Number Problem . Given an integer N, can we find a pair of non-trivial integers x,y such that both  x² + y²   and   x² + N y²  are integer squares.

 
2. A similar problem to the concordant number problem. Given an integer N, can we find a pair of non-trivial integers x,y such that both  x² + N y²   and   x² + (N+1) y²    are integer squares.

 
3. Given an integer N, can we find a pair of non-trivial integers x,y such that both  x² + N x y +  y²   and   x² - N x y + y²    are integer squares.

 
4. Given an integer N, can we find a pair of non-trivial integers x,y such that both  x² + N x y + N²  y²   and  N ² x² + N x y + y²    are integer squares.

 
5. Given an integer m, can we find non-trivial integers x and y with  x⁴ + m x ² y²  +  y⁴   an integer square.

 
6. Given an integer N, can we find a pair of non-trivial integers x,y such that both  x² + N y²   and   x² + N²  y²    are integer squares.

 
7. Given an integer N, can we find a pair of non-trivial integers x,y such that both  x² + N y²   and   x² + y² / N   are integer squares.

 
8. Given an integer N, can we find non-trivial integers x,y such that

x⁴ + N y⁴ is an integer square.
 

   These are elliptic curves given directly with no seeming relation to an underlying problem.
 

1. DATE Curves.  These are the curves y^2 = ( x - D )( x - M )( x - Y ) where D/M/Y is a date - 25 December 2009 gives y^2=(x-25)(x-12)(x-2009).

5  Classic Problems (none written yet!)

1. The integer cuboid problem, where the edges, side diagonals and space diagonal are to be integers.

 
2. Find a point inside an equilateral triangle with integer sides, with the point being at integer distance from the vertices.

 
3. Find a point inside or on an integer square such that the distance from this point to each vertex is an integer square.

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.