What is the smallest number that can be expressed as the sum of two positive cubes in two different ways?

The story behind the answer

1729 is the smallest number expressible as the sum of two positive cubes in two different ways: 1729 = 1^3 + 12^3 = 9^3 + 10^3. No smaller number has two distinct pairs of positive integers whose cubes sum to the same value, which is why 1729 holds this distinction. In number theory, it’s known as the first taxicab number, Taxicab(2).

This is the famous Hardy–Ramanujan number. When G. H. Hardy visited Srinivasa Ramanujan in a taxi numbered 1729, he remarked it seemed dull. Ramanujan replied that it was very interesting for exactly this two-cubes property. To remember: picture a taxi with plate 1729 “carrying” two cube pairs—(1, 12) and (9, 10).