This is a direct result of Wiles (1994) proof of Fermats Last Theorem.

Now there are also a different class of assertions that cannot be proved either way. (Godels Incompleteness Theorem)
I am digressing a lot here, but if you find this sort of thing interesting, i suggest reading 'Godel, Escher, Bach' by Hofsteadter (spelling?)
...prepare to hurt your brain :)

For example, there are no positive integers {a,b,c} such that a^3 + b^3 = c^3

This is a direct result of Wiles (1994) proof of Fermats Last Theorem.

No, this case is much easier to prove than the general case, and was proven in the 18th century by Euler and Legendre.

In fact, for all sufficently small n exponents the impossibility of a^n + b^n = c^n was proven long ago, in fact, Szalay[1] which was published in 1991 reports all n < 125000.

[1] Dr. Szalay Mihály, Számelmélet. Tankönyvkiadó, Budapest, 1991

The fact that there is an easier proof of the special case doesn't change the fact that the special case is a direct result of the general case.