Fermat's Last Theorem states that the equation
x^
n +
y^
n =
z^
n has no solutions in integers with
n>2 and none of the variables 0. It was conjectured by Pierre de Fermat, but probably not proved by him (though he thought he'd proved it).
PierreDeFermat scribbled it on a book's margin:

 Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caparet.

 There are no positive integers such that x^n + y^n = z^n for n>2. I've found a remarkable proof of this fact, but there is not enough space in the margin [of the book] to write it.
(The most likely scenario is Fermat wrote this, thought of an error with his proof in his head, never wrote it down, never bothered to retract the marginalia for obvious reasons, and died. In fact he later in life was still working with n=4.)
FLT was proved in the closing years of the 20th century after extensive work by a number of mathematicians, notably Gerhard Frey, Ken Ribet, JeanPierre Serre, Andrew Wiles and Richard Taylor. Wiles's name is the one most usually attached to the proof, because he published the first version of the last piece in the puzzle. (I think his contribution was the deepest, too.)
Wiles's work has led to a proof of the socalled
TaniyamaShimuraWeilConjecture, which was one of the most important unsolved problems in number theory. (This was always Wiles's ultimate aim, though Wiles is not among the authors of the paper in which TSW is proved, which is due to appear in the
Journal of the AMS soon.)
Before FLT was properly proved, many attacks had been made on it. It was proved to be true for many particular values of
n (some of the proofs being very elegant); for values of
n with particular properties; and (most recently) true with at most finitely many exceptions for all
n [The
Mordell Conjecture, that any algebraic curve of genus > 1 has only finitely many solutions in any algebraic number field]. (That one was done by Gerd Faltings, again as an offshoot of more general work.)
And, of course, there were many, many,
many bogus pseudosolutions from crackpots. :)
You'll find some every single time you click here: news:sci.math.
 Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem (ISBN 0385493622 )
 ... a documentary of Wiles's work (but lacks mathematical depth (assumed too hard for readers to follow)).
 Notes on Fermat's Last theorem, by A. J. van der Poorten (ISBN 0471062618 )
 Fermat's Last Theorem, by Simon Singh (ISBN 1857026691 ) is a good read, but also lacks depth.
It was moving to see the television documentary on Wiles quest when he recalled the moment of discovery and was almost moved to tears in the interview. An example of a
LifeGoal achieved. Also illustrates that mathematics is very much an emotional and aesthetic endeavour not a just a "mechanical" one.
Does anyone know if the actual book
PierreDeFermat wrote his note in, Diophantus'
Arithmetica still exists (is it viewable in a museum somewhere for instance?). I know his son collected his papers and the book after his death and published a special edition with the note printed in, but is his own copy with handwritten note still intact? I would think that would be a historical treasure. I can't seem to find any reference or images of the original.
No, it doesn't still exist.
[If his son recognized its importance to the point he got published a special edition, why was it not preserved? What were the circumstances of its loss? It's like saying
LeonardoDaVinci once painted a lady with an enigmatic smile but no one can find it (in fact Mona Lisa used to have 2 columns at each side that were cut off, so art and artifacts do get destroyed and lost. But I am curious as to what happened to Fermat's book). After all they preserved Galois' notes on Group Theory the night of his death ("I have not time...") ironically Galois' constructs were a major part in the proof of FLT]
In my day, sonny, FermatsLastTheorem was just a conjecture.
I appreciate that this was meant humorously, but for those who were wondering, it's always been referred to as FLT because Fermat claimed he had proved it.
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