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On another thread someone posted a problem from their math homework: to prove that 1729 is the smallest non-trivial taxicab number (or, if you prefer, that $Ta(2)=1729$). Commenters suggested simply brute forcing it, which I take as an indication that there's no elementary proof. But do we know how Ramanujan knew? Did we brute force it before the advent of computers?

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kuzzooroo
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    We do not know how Ramanujan knew. He discovered mathematical results which seem impossible to find without modern theory or computers. The question is now answered here, – Dietrich Burde Jun 11 '15 at 19:15

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I think brute force was probably the way to go, once you had noticed that 1729 had the desired property. You would only need to check at most 12 x 12 / 2 = 72 sums as floor (cube root 1729) is 12.

sjb
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