From: buzzard@math.berkeley.edu (Kevin Buzzard)
Newsgroups: sci.math.numberthy
Subject: Integer points on elliptic curves.
Date: 24 Apr 97 13:07:25 GMT

With reference to my recent posting about finding integer points
on elliptic curves.

Thanks for all your replies. Just a quick summary, because the
results were rather interesting.

Firstly, several people told me that the package simath would
find integer points on elliptic curves. Someone else told
me that the newest version of Kant would do it too. We don't
have either of those things on our system here, but I was
fortunate enough that for both packages someone took the trouble
to run Y^2=X^3+17 through for me and send me the output.

Bjorn Poonen took a different tack, he remarked that the
curve Y^2=X^3+17 was in fact dealt with in Silverman's
first book on elliptic curves. I am very grateful
to him for this remark, because without it I perhaps may
not have realised that the outputs from both packages
were wrong! Neither package found the point [5234, 378661]
and in fact they both missed several others. According
to Silverman, there are 16 integral points on the
curve. I'm afraid I can't tell the relevant people
which versions of which programs were being used,
and I haven't verified the calulations myself,
but I did find this rather interesting. The curve has
rank 2, by the way. I also don't know whether the programs
do claim to find all integral points, although I certainly
got the impression from the outputs sent to me that the programs
both thought that they had done it, and I also was vaguely under
the impression that computers nowadays were perfectly capable
of doing Y^2=X^3+17 correctly.

Finally, N. Bruin sent me email saying that I had a dangerous
attitude, just asking for packages instead of methods, because
packages sometimes make mistakes. An interesting comment,
in hindsight. I should perhaps just mention that this whole thing
was nothing to do with my research, I was in fact giving a talk
to a general scientific audience and I wanted to mention how
hard it was to solve equations in integers in general, I thought
I'd give an elliptic curve as an example. Turned out it was
harder than I thought :)

Kevin

==============================================================================

From: andreas@cs.umanitoba.ca (Andreas Stein)
Newsgroups: sci.math.numberthy
Subject: Re: Integer points on elliptic curves
Date: 24 Apr 97 16:08:57 GMT

With reference to the recent posting of Mister Kevin Buzzard
on finding the integer points on elliptic curves.

I want to let everybody know that the informations Kevin Buzzard
submitted to the number theory net were not quite correct.

The computation of integer points of the elliptic curve
Y^2 = X^3 + 17
is a simple problem and was performed correctly by SIMATH.
The result is, as one easily checks,

      1. ( -2 , 3 ) =  ( 0  1  0 )
      2. ( 8 , 23 ) =  ( 0  -2  0 )
      3. ( 2 , 5 ) =  ( 0  1  -1 )
      4. ( 4 , 9 ) =  ( 0  0  1 )
      5. ( -1 , 4 ) =  ( 0  -1  -1 )
      6. ( 52 , 375 ) =  ( 0  2  1 )
      7. ( 43 , 282 ) =  ( 0  1  -2 )
      8. ( 5234 , 378661 ) =  ( 0  -1  -3 )

where in this representation the right sides denote the linear combinations
of the points with respect to a basis computed by SIMATH.
Of course, the 8 missing points are the inverses of the 8 mentioned points
meaning that there are 16 integral points on the curve.

When mentioning a particular computer algebra system in a message,
the author should be aware of the correctness of his statements.
Either he or one of his members have no experience in using
computer algebra systems (by the way KANT also gives the same answer)
or they only tried out one system and generalized the wrong result.

In particular, for these kinds of questions SIMATH is well-known
and acknowledged.

Andreas Stein.