Jeff Miller wrote:
| Harold Edwards writes on page 2 of "Fermat's Last Theorem: A Genetic
| Introduction to Algebraic Number Theory" (Springer, 1977), "The origin
| of the name 'Fermat's Last Theorem' is obscure."
As I already said, the origin of the name 'Fermat's Last Theorem' is not
obscure at all.
| "Very possibly --remarks H. Edwards -- the name stems from the fact
| that of the many unproved theorems that Fermat stated, this is the
| last one that remains unproved."
Gabriel Lame makes it very clear that:
"De tous les theoremes sur les nombres, enonces par Fermat,
un seul reste incompletement demontre."
[Of all the theorems on numbers stated by Fermat, just one remains
incompletely demonstrated (proved)]
"Ce theoreme dit que l'equation x^n + y^n = z^n est impossible
en nombres entiers, lorsque l'exposant n est plus grand que 2."
[This theorem states that the equation x^n + y^n = z^n is impossible in
integers (x, y, and z, all different from zero, JGC), when the exponent
n is greater than 2. (And so this is the *last* theorem that remains
unproved, JGC)]
As you may gather from Lame's wording, there is no trace of doubt as to
the validity of FLT. Lame is assuming from beginning to end that FLT is
a theorem (and not a conjecture!). It's rather late here in Montevideo,
and I hope my ideas are being properly conveyed. If not, I would love to
write an entire page in my native language, Spanish (or better, Uruguayan ;-)
In his _Rapport sur un memoire de M. Lame", Cauchy remarks (*):
"L'Academie nous a charges, M. Liouville et moi, de lui rendre
compte d'un Memoire de M. Lame sur le dernier theoreme de Fermat."
[The Academy has charged (asked) us, Mr Liouville and myself, to review
(give account of) a memoir of Mr. Lame on the last theorem of Fermat.]
"On sait que Fermat, l'un des plus beaux genies qui aient illustre
la France, a donne des enonces de plusieurs theoremes, parmi
lesquels il en est deux dont la demonstration a ete pendant
long-temps recherchee avec ardeur par divers geometres. De ces
theoremes il n'en reste plus qu'un seul qui ne soit pas aujourd'hui
completement demontre : c'est le theoreme relatif aux puissances
des nombres entiers, et suivant lequel une puissance d'un
degre $n$ superieur au second, ne peut resulter de l'addition de
deux puissances du meme degre." [p. 359]
Again, in this passage, Cauchy states very clearly that of the many
unproved theorems that Fermat stated, the FLT is the *last* one that
remains unproved ["De ces theoremes il n'en reste plus qu'un seul qui
ne soit pas aujourd'hui completement demontre"]. So, I am very sorry
to say that there is *nothing* very obscure here.
[*] C.R. Acad. Sci. Paris, 9, 1839, pp. 359-363.
An early reference to the term "Last Theorem of Fermat" in the English
language appears in "Application to the Last Theorem of Fermat" (1860),
in "Report on the Theory of Numbers", part II, art. 61, addressed by
Henry J.S. Smith.
With my best regards to all, as always,
Julio Gonzalez Cabillon