Ramanujan, Hardy and the Number 1729

The mathematician Godfrey Harold Hardy (G. H. Hardy, no confundir con Oliver) tells the story of visiting the Indian mathematician Srinivasa Ramanujan who was ill at the time. As Hardy mentions, "I had ridden in taxi-cab No. 1729, and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen.

- 'No, he replied, 'it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways'.

What are the sums of which Ramanajuan speaks?

Note: The numbers in this problem are all positive integers.

Extracted from The World of Mathematics, Volume 1, by James R. Newman, p. 375.