Achilles: Well, in the mathematics of acoustico-retrieval, there arise many
questions which have to do with the number of solutions of certain
Diophantine equations. Now Mr. T has been for years trying to find a
way of reconstructing the sounds of Bach playing his harpsichord,
which took place over two hundred years ago, from calculations involv-
ing the motions of all the molecules in the atmosphere at the present
time.
Anteater: Surely that is impossible! They are irretrievably gone, gone
forever!
Achilles: Thus think the naive ... But Mr. T has devoted many years to
this problem, and came to the realization that the whole thing hinged
on the number of solutions to the equation
an + bn = en
in positive integers, with n > 2.
Tortoise: I could explain, of course, just how this equation arises, but I'm
sure it would bore you.
Achilles: It turned out that acoustico-retrieval theory predicts that the
Bach sounds can be retrieved from the motion of all the molecules in
the atmosphere, provided that EITHER there exists at least one solution
to the equation-
Crab: Amazing!
Anteater: Fantastic!
Tortoise: Who would have thought!
Achilles: I was about to say, "provided that there exists EITHER such a
solution OR a proof that there are NO solutions!" And therefore, Mr. T,
in careful fashion, set about working at both ends of the problem,
simultaneously. As it turns out, the discovery of the counterexample
was the key ingredient to finding the proof, so the one led directly to
the other.
Crab: How could that be?
Tortoise: Well, you see, I had shown that the structural layout of any proof
of Fermat's Last Theorem-if one existed-could be described by an
elegant formula, which, it so happened, depended on the values of a
solution to a certain equation. When I found this second equation, to
my surprise it turned out to be the Fermat equation. An amusing
accidental relationship between form and content. So when I found
the counterexample, all I needed to do was to use those numbers as a
blueprint for constructing my proof that there were no solutions to the
equation. Remarkably simple, when you think about it. I can't imagine
why no one had ever found the result before.
Achilles: As a result of this unanticipatedly rich mathematical success,
Mr. T was able to carry out the acoustico-retrieval which he had so
long dreamed of. And Mr. Crab's present here represents a palpable
realization of all this abstract work.Prelude ... 279