Much enjoyment and a little sleep deprivation with this problem, but I think it's now done..
I felt sure there was a way of proving that the two solutions already posted were the only ones, but having tried modular arithmetic etc. I began to realise it was more about 'elliptic' curve theory..(or is there a more obvious approach?).
Starting with 2s³ = 3t² + 4, multiply by 4 to get 8s³ = 12t² + 16, then write the right hand side as the sum of two cubic functions with terms that cancel each other out:
(2s)³ = (2 + t)³ + (2 - t)³
Now, Fermat/Wiles says that this equation has no integer solutions except the trivial ones: t=2, s=2 and t=-2, s=2, which are the ones posted earlier.
Edited on June 18, 2009, 12:50 am
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Posted by Harry
on 2009-06-17 23:15:49 |
Analytical solution
