Since Fermat had posed his problem, Euler was the
first to make the first big progress. He proved the individual case n=3, in the year 1770,
using the technique developed by Fermat himself which is called infinite descent. (please click for
the details of the proof) Indeed Fermat used this method in proving the 
case when n=4. The reasoning is not difficult. The essence of this method
is to prove that if there is positive integer solution for the equation, then smaller
solutions in integer with exactly the same properties can be derived from the first
solution and so on for infinite times. But for a positive integer, it is impossible for it
to decrease indefinitely. Hence a contradiction is set up and a proof that there is no
solution for the equation is established.
Pierre de Fermat (1601-1665)
Euler
adopted Fermat’s approach while adding into his proof complex number to establish his
reasoning. However, his proof was flawed, though it was not a fatal mistake. In short, he
has assumed that if a number = (a+b*-31/2 )(a-b*-31/2)= a cube where
a and b are integers relatively prime and (a+b*-31/2) and (a-b*-31/2)
are then as a result "relatively prime", then each of which (a+b*-31/2)
and (a-b*-31/2) is
also a cube. He has assumed the same properties for irrational complex number as integers
without any solid proof.
Leonhard Euler(1707-1783)
Fortunately, his proof
was fixed by Legendre. In the following ninety years after Euler, there 
were not many great breakthroughs. The most important of all is Sophie
Germain’s Theorem which directly led to the proof of the case n=5 given by Dirichlet
and Legendre. The approach intends to proof that for n=p where p is a prime and 2p+1 is
also a prime(p is called Germain Primes), then one of the three solutions, x, y or z must
be divisible by n. Through this theorem, any proofs for cases that n= Germain Primes only
need to show that for all solutions x, y and z are not divisible by n.
Adrien-Marie Legendre (1752-1833)
During 1847, there
was an important event in the history of proving Fermat’s Last Theorem. On 1 March in
a meeting of the Paris Academy, Lamé proudly announced that he had found a proof for the
Theorem finally, though not having completed that yet. However, his proposed 
proof seemed very promising. The proofs for cases like n=3,4,5,7 given
before relied heavily on some algebraic factorization such as x3+y3=
(x+y)(x2-xy+y2) in the case n=3. However, this method gets
increasingly difficult as n increases since one of the factors in this decomposition has
very large degree. Lamé thought that this could be overcome by decomposing xn+yn
completely into n linear factors. This is done by introducing a complex number r where rn=1
(r=cos(2p /n)+isin(2p /n)) and using the algebraic identity:
Gabriel Lamé (1795-1870)
xn+yn=(x+y)(x+ry)(x+r2y)…(x+rn-1y) (where n is an odd integer)
If he could show that if x and y are such that the factors (x+y), (x+ry) (x+r2y), …, (x+rn-1y) are relatively prime, then each of them must be an nth power and an impossible infinite descent can then be derived from.
This is really ingenious, but there are some gaps in the reasoning, pointed out by Louville, who had contributed the idea to Lamé in a casual conversation. He asked how Lamé could establish the validity of his conclusion that each factor was an nth power if all he had shown was that the factors were relatively prime and that their product was an nth power which is true for ordinary integers but uncertain in the case for complex numbers. At the same time, Cauchy, a contemporary of Lamé also saw the possibility of solving the Theorem using a similar method and he too claimed to be completing the final proof. Both of them didn't accept defeat until they received the devastating letter from Kummer who proved that their proofs must fail. Lamé felt silent but Cauchy wouldn't give up. He continued to publish vague and inconclusive articles on the proof. But by the end of the summer, he too fell silent.
Kummer, though refuted Lamé' s proof,
introduced a new kind of complex numbers which he called "ideal complex numbers"
and led to the resurrection of the theory of factorization. By applying his new theory to
Fermat last theorem, he could reduce its proof for a given n to
the testing of
two conditions. Using this technique, he found signs that seemed to indicate that the case
n=37 do not satisfy the Theorem.
Basing on his theory, Kummer did not invalidate Fermat last theorem. In contrast, he gave a proof on all cases where n= regular primes , a group of numbers he defined to satisfy 2 particular conditions. This was the greatest progress made since the problem was posed.
Ernst Eduard Kummer (1810-1893)
From then on, not much developments were made. It was finally proved not until there were the right mathematical tools developed by Andrew Wiles.