How many triplets of positive integers (a,b,c) satisfy a^6+b^6+c^6=140000004.
I have no idea. I know that the 3 cubes problem is unresolved, and this is roughly equivalent to finding the sum of three cubes, or even three squares. Still, even a brute force approach would not take too long on a computer, since ∛140000004 ≈ 519, so at most 519^3 ≈ 140 million triples would have to be tested, and in practice, much fewer than that as the sums are found to exceed the desired value.
Any number theorists out there? I had one class in number theory, way back when, but I don't recall discussing this or any similar problems. Better try google first, I guess.