16 January 2009: Have a Good New Year!! I have decided to introduce a new section - on Recreational Elliptic Curves. These are elliptic curves which do not come from an underlying problem, but are studied for themselves. The first example are what I call DATE curves  y^2 = (x-D)(x-M)(x-Y) where D/M/Y is a date, so today's curve is y^2=(x-16)(x-1)(x-2009) which has rank 0.
 

10 September 2008: I have been doing some work on the classic problem of rational solutions to x^3 + y^3 = N, for N cubefree. I have found solutions for N in the range [1,9999].

25 August 2008: I have only recently returned to work after being seriously ill for over a year. I am attemting to get my work-life back in order and hopefully update some of these pages.

28 November 2006:  The problem  of x^2 + N x y + y^2 and x^2 - N x y + y^2 both being squares has now been resolved for all N in [1,999]. The N=809 solution was found by a combined effort from Tom Fisher, Mark Watkins and Randall Rathbun. There are no immediate plans to extend this range.

21 July 2006:  Finished off  ALL the unsolved problems in the quartic x^4 + N x^2 y^2 + y^4 = a^2 problem. This was only possible using 2 labs of PCs which were unused as it is the student vacation. Without this, the task would have been really tedious.

7 June 2006:  Found a solution for N = -3675 problem in the quartic x^4 + N x^2 y^2 + y^4 = a^2 problem. This had height 306, but has an isogeny with height 153. I used Heegner points linked to four-descent. The computation was only practical as I persuaded my institution to give me access to a student lab of PCs during the vacation.

6 June 2006:  Introduced a new page discussing Congruent Numbers.

2 June 2006:  I have been doing some work on the Congruent Number problem and am about to add a page. I have also started investigating solutions to another classic -  x^3 + y^3 = N z^3.

2 June 2006:   I have discovered the N = -3615 solution in the quartic problem from the pages  x^4 + N x^2 y^2 + y^4 = a^2  which had height 219. This involved a mixture of isogeny finding, four-descent and Heegner points, together with running a Pari code over the weekend on 2 old machines.

8 May 2006: The problem  N = x / y  +  y / z  +  z / x has been completed for N in the range  [-199,-1].

4 May 2006:  I have done some work on the equation x^4 + N y^4 = z^2  and have produced ALL solutions for N in [-9999,9999].

26 April 2006:  I have started the (laborious) process of updating the symbol representation on these pages, since they were becoming unreadable on some web-browsers. Thanks go to Hugo van der Sanden for showing me how to do this. There are probably still problems since I don't have access to all possible browsers, only Internet Explorer, Netscape and Mozilla Firefox.
 

21 April 2006:  The problem  of x^2 + N x y + y^2 and x^2 - N x y + y^2 both being squares has now only 1 value in [1,999] left to resolve. With help from Randall Rathbun, I found the N=679 solution.
 

6  March  2006:  The x^4 + N x^2 y^2 + y^4 = a^2  problem is now almost complete for [-9999,9999], due mainly to the heroic efforts of Randall Rathbun with Tom Womack chipping in. A table of remaining values is now available.
 

4 January 2006:  I returned from the Xmas break to find Randall Rathbun had sent me some more large height solutions in the problem  of  x^2 + N x y + y^2 and x^2 - N x y + y^2 both being squares. There are now only 2 values in [1,999] left to resolve.
 

26 November 2005:  Randall Rathbun recently sent me several large height solutions in the problem  of  x^2 + N x y + y^2 and x^2 - N x y + y^2 both being squares. There are now only 7 values in [1,999] left to resolve.
 

22 August 2005:  I managed to find all the remaining unsolved solutions to the generalised congruent number problem.
 

25 July  2005: Randall Rathbun sent a file with solutions for the concordant   number problem with negative N. There are 39 unsolved values, with large heights and hence very large solutions.
 

10  May  2005:  The x^4 + N x^2 y^2 + y^4 = a^2  problem is now complete for all POSITIVE values up to 9999. The final few were finished mainly by Randall Rathbun, and one large Heegner calculation by myself.
 

26 April 2005:  Leech N=502 found by Randall Rathbun and so all solutions in [1,999] have been found (assuming BSD conjecture).
 

25 April 2005:  Randall Rathbun also sent me a large number of solutions for
the x^4 + N x^2 y^2 + y^4 = a^2  problem. I have updated the results files.
The solutions are now complete in 4 of the ten intervals, and there are only 9 positive N values left.
 

23 March 2005: Further computations on the x^4 + N x^2 y^2 + y^4 = a^2 problem have found all solutions for 1 <= N <= 5999 and all bar one solution for -1999 <= N <= -1. The one outstanding value is N = -1875.
 

22 March 2005: The problem  of  x^2 + N y^2 and x^2 + (N+1) y^2 both being squares has been completed for N in [1,999].
 

17 March 2005: For Leech's problem, I have reduced the number of unsolved N values to 1, namely N=502. I would quite like to get rid of this one - any volunteers? See above 26/04/2005 news
 

16 March 2005: Knight's problem finally solved for original range of
[-999,999]. Thanks go to Mark Watkins of the Magma team for the final 4 solutions.
 

9 March 2005: Finally managed to get the

N = ( x^3 + y^3 + z^3 ) / ( x + y + z )^3

              page loaded.
 

1 March 2005: I was recently sent several solutions to the quartic problem

             z^2 = x^4 + N x^2 y^2 + y^4

by Randall Rathbun and was inspired to find some further solutions. These can be found here. There are now only 39 unsolved N values for 1 <= N < 9999!
 

28 February 2005: One of the few remaining unsolved values for Knight's problem has been resolved by a somewhat lengthy Heegner point computation.

The solutions for 699 = ( x + y + z ) ( 1/x + 1/y + 1/z ) are
 

x = 183314213070737187395144070401992080881832872345699
    656656954404713619153495035523439505984256202601192
    07264808469542958137261248076668202149667286599515

y = 240566534149017393075296472432248945218365199493346
    989437999267750085550355904147420589802549869756708
    33083700452921718959257257700992683787674356528

z = -12263402851354724639593377170468333121830925170434
    46298351555346136922843263408398678709253774931501
    688370280913316769768948953890023412941261248898683
 

10 January 2005:The concordant number problem has been resolved for positive values of N up to 9999. Thanks to Randall Rathbun for several large height solutions which finished off the unknowns.