It is well known that Euler gave the first proof of FLT ($x^n + y^n = z^n$ has no nontrivial integral solutions for $n > 2$) for exponent $n=3$, but that his proof had gaps (which are not as easily closed as Weil seems to suggest in his excellent Number Theory - An Approach through History). Later proofs by Legendre and Kausler had the same gap, and in fact I do not know any correct proof published before Kummer's proof for all regular primes. Gauss had a beautiful proof, with the 3-isogeny clearly visible, which was published posthumously by Dedekind, and of course Dirichlet could have given a correct proof (he gave one for $n = 5$ in his very first article but apparently did not dare to provoke Legendre by suggesting his proof in Theorie des Nombres was incomplete) but did not.

The problem in the early proofs is this: if $p^2 + 3q^2 = z^3$, one has to show that $p$ and $q$ can be read off from $p + q \sqrt{-3} = (a + b\sqrt{-3})^3$. The standard proofs use unique factorization in ${\mathbb Z}[\zeta_3]$ or the equivalent fact that there is one class of binary quadratic forms with discriminant $-3$; Weil uses a (sophisticated, but elementary) counting argument.

I wonder whether there is any correct proof for the cubic Fermat equation before Kummer's proof for all regular prime exponents (1847-1850)?