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
