Proof that $1729$ is the smallest taxicab number

For homework I have to produce the proof (algebraic or otherwise) to show that $1729$ HAS to be the smallest taxi cab number. A taxicab number means that it is the sum of two different cubes and can be made with $2$ sets of numbers. I have the list of the next ones and I was wondering if it was linked with the fact that it would have to be $0$ cubed if it got any lower which obviously wouldn't work.

Any help appreciated, thanks in advance!

One can prove that the smallest taxicab number is the smallest product $(6n+1)(12n+1)(18n+1)$ consisting of three primes. This means $n=1$, and $7\cdot 13\cdot 19=1729$. I do not claim that this proof is much better than brute-force.