It is known that there is no solution in N to

x^5 + y^5 = z^5 (FLT),

but there is a solution to

x^5 + y^5 + u^5 + v^5 = w^5

This was the first counterexample to Euler's conjecture.

What about sum of three fifth powers

x^5 + y^5 + z^5 = t^5

Does a solution exist? Could someone please say something about this case? Thank you.

a priorithat there would be infinitely many primitive solutions), but we have no techniques to prove such a result. I might post an answer later detailing why, but have some things to tend to first before dinner... – Noam D. Elkies Aug 9 '11 at 18:15