Apparently, finding an integral solution (integers being positive or negative numbers without a fractional component) for this little equation is harder than one might think.
According to the internet (and I am taking the internet’s word for it here) x3 + y3 + z3 = n is full of surprises. For example…
For this equation:
x3 + y3 + z3 = 29
a solution is easy to find: (x,y,z)=(3,1,1). Whee!
When n=30 it is harder to find a solution, but one is known:
When n=33 it is expected that there is a solution but none is currently known. Calling all Amazon Mechanical Turks! Let’s figure this one out so I can post the answer on my blog, okay? For anyone who is interested, here is a chart of solutions of x3 + y3 + z3 = n when n is a whole number less than 100.