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26: x^{3} + y^{3} + z^{3} = 33

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) x^{3} + y^{3} + z^{3} = n is full of surprises. For example…

For this equation:

x^{3} + y^{3} + z^{3} = 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:

(x,y,z)=(2220422932,−2218888517,−283059965). Whoa.

And….

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 x^{3} + y^{3} + z^{3} = n when n is a whole number less than 100.