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 | | From: | John Robertson | | Subject: | Re: Diofrantic | | Date: | Sun, 23 Jan 2005 23:53:12 +0000 (UTC) |
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 | On 22 Jan 2005, Ian Hutcheson wrote:
>I would be
>interested if someone could select a simple example, (or
>reference), to demonstrate the general idea of the "method
>of descent", for
>
>a^3 +b^3 = K*c^3
>
I don't know about ``descent'' methods, but the first thing to note
is that a^3 + b^3 = K*c^3 is an elliptic curve. J. W. S. Cassels,
``Lectures on Elliptic Curves,'' London Mathematical Society Student
Texts, 24, Cambridge University Press, Chapter 8, Section (i), page
34, shows how to reduce the curve to Weierstrass form. From this
solutions can be found from methods in John Cremona's book
``Algorithms for Modular Elliptic Curves'' and or using his programs
(see http://www.maths.nott.ac.uk/personal/jec/ftp/data/INDEX.html),
or using The APECS package for Maple by Ian Connell
(http://www.math.mcgill.ca/connell/). See also, L. J. Mordell,
``Diophantine Equations,'' Academic Press, 1969, pages 124 to 130.
John Robertson
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