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256 CHAPTER 22
if r = 3. (This last curve does not look as if it ﬁts our deﬁnition of
an elliptic curve, but a certain change of variables shows that it is
one, in fact.) But if r > 3, elliptic curves do not seem to work, and
more complicated varieties need to be employed.
Prospects for Solving the Generalized Fermat Equations
Darmon has gone on to sketch a method to solve other generalized
Fermat equations but the various conjectures needed to make this
method work have not yet been proven. It seems that some new
ingredients will be needed. Indeed, other people are carrying on
the work with various new ingredients. Here is a quotation from an
e-mail message we received from Darmon on February 9, 2005:
I had the impression ...that further progress would have to
wait for someone more clever, or bringing in additional tools,
to come along. Fortunately, people like that did step in. There’s
been some nice work on generalised Fermat equations by Alain
Kraus ....
Bennett and Skinner have also done some interesting work,
and Ellenberg as well [see chapter 23] ....
There has also been very interesting work of Bugeaud,
Mignotte, and Siksek combining the modular forms techniques
with more traditional approaches like linear forms in logs
to solve striking open problems, like the complete list of
perfect powers in certain binary recurrence sequences (e.g.,
the Fibonacci and Lucas sequence). I suspect that this is where
the future of the subject lies: Both the methods based on mod-
ular forms, and more traditional approaches, run into serious
obstacles when dealing with natural Diophantine equations,
but because those methods are so different, the obstacles one
encounters are likely to be different, so one can hope that the
information gleaned from a combination of approaches can be
stronger and lead to a solution, where no technique applied by
itself could.

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