Fermat's last theorem

Pierre de Fermat published very little. He communicated most of his results in letters to friends (usually without proof). More of Fermat's results were later discovered written in the margin of his copy of Diophantus' Arithmetica. By far the most famous is the one called Fermat's Last Theorem:

"It is impossible to write a cube as a sum of two cubes, a fourth power as a sum of fourth powers, and, in general, any power beyond the second as a sum of two similar powers. For this, I have found a truly wonderful proof, but the margin is too small to contain it." [This was written by Fermat near 1637, in Latin.]

Now we would write this as follows.

Fermat's Last Theorem: (Wiles 1995)
The equation xⁿ + yⁿ = zⁿ has no solutions in positive integers for n greater than 2

This result is called his last theorem, because it was the last of his claims in the margins to be either proved or disproved. Few (now) believe Fermat had found the proof he claimed. Wiles found the first accepted proof in 1995, some 350 years later. Wiles proof is exceptionally long and difficult!

See Also: BealsConjecture, CatalansProblem

Related pages (outside of this work)

The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles Gerd Faltings summary of the proof (an Adobe Acrobat version)

References:

Ribenboim1979
P. Ribenboim, 13 lectures on Fermat's last theorem, Springer-Verlag, 1979. New York, NY, pp. xvi+302 pp. (1 plate), ISBN 0-387-90432-8. MR 81f:10023
Ribenboim1999
P. Ribenboim, Fermat's last theorem for amateurs, Springer-Verlag, 1999. New York, NY, pp. xiv+407, ISBN 0-387-98508-5. MR 2001h:11036
TW95
R. Taylor and A. Wiles, "Ring-theoretic properties of certain hecke algebras," Math. Ann., 141:3 (1995) 553--572. MR 96d:11072 [The rest of Wiles' proof] [Here Wiles and Taylor fill in the gap which was spotted in the original version of Wiles proof of Fermat's last theorem. The rest of the proof is in [Wiles95].]
Wiles95
A. Wiles, "Modular elliptic curves and Fermat's last theorem," Ann. Math., 141:3 (1995) 443--551. MR 96d:11071 [Wiles' proof] (Annotation available)