>Greetings, and thanks so much for your reply! Please again forgive my
>ignorance of this subject, but is there any significance in
>Diophantine analysis to equations which have a single solution in the
>integers, like this one, as apart from a finite number?
No; the big gap in Diophantine analysis is between "finitely many" and
"infinitely many". Surprisingly, this often is of much greater general
significance than the difference between "some" and "none"!
>Are there
>many/(any other) such equations of which you are aware?
You mean Diophantine equations with exactly one solution?
I don't know of any specific examples, but there are assuredly very
many of them. If you choose a Diophantine equation at random, and
its degree is not too low, there will almost certainly be only a
finite number of solutions; and in many cases the number will be
quite small (0, 1, or 2). I don't know of any work that's been
done on the distribution of the number of solutions of "random
Diophantine equations", but I'd expect 0 to be the most common number
of solutions, followed by 1, followed by 2, etc. (Keep in mind that
number theory is not my field, so "I don't know of any" doesn't mean
"There doesn't exist any".)
>Did Fermat or
>the others following him simply consider this equation as one of many
>special equations to be analyzed, or are the number of equations
>receiving comparable interest relatively few?
Fermat and his successors focussed on specific examples, largely because
they didn't have the tools required to handle infinitely many examples
at once. 25 = 27 - 2 and 8 = 9 - 1 are certainly the most attractive
problems of the "a perfect cube differing from a perfect square by a
designated amount" variety, since the numbers (especially the numbers
1 and 2) are so small. I'm not sure how many other equations of this
type received special treatment back in those days; you could look
in the chapter of Dickson entitled "Binary Quadratic Form Made an
Nth Power".
Nowadays, many (if not most) of the top number theorists are more
interested in general theories than individual problems. They do
acknowledge that the test of a good general theory is that it should
be capable of being applied to individual problems. But a great
many number-theorists have comparatively little interest in these
applications.
Jim Propp
Department of Mathematics
University of Wisconsin