How many different solutions are there to the equation x^3 + y^3 = z^3?
The number of possible solutions to this equation depends on whether you consider real, complex or rational numbers.

For real numbers we have:
(x ± y)^3 = (z ± w)^3
for all choices of signs x, y and z as well as for w = 0
So in total there are 6 possible solutions.

For complex numbers there is a similar result but with more restrictions:
(x +iy)(x - iy) = (z +zw')i (z - zw')j
with j being the imaginary unit and i² = -1
So again there are 6 solutions.

For rational numbers there is no general solution. But if we assume that one of the three variables is a perfect square then we can solve the equation. This leads to 12 solutions when we only allow squares up to 1, 2 or 4. For larger values the number of solutions grows quickly so it's not feasible to list them all here.