Fermat put forth his 'last theorem' in 1637 which says that
No two positive integers greater than two satisfy the equation
$a^n + b^n = c^n$
But Fermat claims that the margin in his notebook was not big enough to accommodate the large proof. Which simply made it a conjecture that was never to become a theorem until after 358 years of mathematicians attempting to prove it.
The first successful proof was published in 1995 by Andrew Wiles who used modern methods such as the modularity theorem which is a thing of the 1950's and Ribet's theorem from 1985.
My question is is there any way by which Fermat could have possibly known the proof for his theorem? Although Andrew Wiles used modern methods, there are many ways to prove a theorem.
The answer probably is that Fermat could have never possibly proved it with the methods he had. (Euler could not provide a general proof either) Now Mathematics still was extremely advanced at times such as that of Fermat and Euler and the theorem even slipped through Gauss, Lagrange, Legendre, Riemann and Dirichlet (all leading proponents of Number Theory). What is it that was lacking that postponed the proof to after three centuries of masters?