What sort of attempts have been made to reconstruct his invalid proofs of these
two assertions?
Fermat didn't give us much to go on, so such reconstructions must inevitably
rely largely on imagination.
Let's discuss FLT first.
One common suggestion is that Fermat used methods that were tantamount to
the consideration of cyclotomic fields, and made Lame's mistake of assuming
unique factorization in those fields. It is certainly true that any argument
that can be given in terms of an algebraic extension of Q can be paraphrased
in terms of Q (and indeed, Gauss' theory of quadratic forms, in all its
difficulty, can be viewed at least in part as just such a rewriting of the
theory of imaginary quadratic extensions). But I still find it hard to see
how one might rewrite Lame's fallacious proof in terms that Fermat would
have understood, _and_ in a fashion that might have tempted him to assume
unique factorization. (It seems to me that the whole temptation to assume
unique factorization hinges on the analogy between arithmetic in Q and
arithmetic in extension fields; if one rephrases arithmetic in extension
fields in tortuous ways that don't involve complex numbers, one would also
be removing the intuitions that would lead one to falsely assume unique
factorization in the first place!) Has anyone addressed this issue?
A suggestion I find more plausible is Bombieri's notion that Fermat might
have found a proof by descent that is actually a circular argument in
disguise. (Bombieri gives a specific argument of this kind, based on
Lexell's correct observation that FLT is equivalent to a certain other
Diophantine equation; if one does the two directions of implication in
series, it's possible one could think that the new solution to FLT is
smaller than the old, instead of being identical to it.)
Are there other reconstructions of Fermat's prof of FLT that have been
offered?
Moving on to Fermat numbers, Bombieri recently offered a suggestion for
Fermat's false primality proof, in which Fermat mistakenly assumes a kind
of converse to his Little Theorem. Have other reconstructions been
proposed?
Jim Propp
Department of Mathematics
University of Wisconsin