"On the other hand, it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power except a square into two powers with the same exponent. I have discovered a truly marvelous proof of this, which however, the margin is not large enough to contain."In other words, Fermat was asserting that if x^n+y^n=z^n for integers x,y,z and n>2, then xyz=0. Since all other statements that Fermat made without proof were either proved or disproved by 1800, this became his Last Theorem.Pierre de Fermat, circa 1636
The problem motivated much of the development of algebraic number theory centered on the study of the fields generated by the roots of unity. By studying unique factorization in certain rings Kummer was able to show that Fermat's conjecture was true in some cases about 1850.
About 1985 Frey observed that a counterexample to Fermat's Last Theorem would give rise to an elliptic curve which had such special properties that its existence would seem to contradict some deep conjectures. For the first time Fermat's conjecture was seen to be a consequence of a deep structural result that most number theorists believed true. Ribet was able to prove that Frey's viewpoint was correct, and thus reduced Fermat's conjecture (and several other Diophantine equations) to an understanding of elliptic curves.
In summer of 1993 Wiles announced that he had proved the necessary results about elliptic curves. There were some gaps to be repaired, but by October 1994 a complete proof was available. As a corollary to these developments we have a proof of Fermat's Last Theorem. More importantly we have made deep progress in understanding elliptic curves.