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

The smallest number that can be expressed as the sum of two positive cubes in two different ways is 1729. This number is also known as the Hardy-Ramanujan number. It was discovered when the mathematician G.H. Hardy visited the Indian mathematician Srinivasa Ramanujan in the hospital. Hardy mentioned that he traveled in a taxi with the number 1729, which he thought was a dull number. Ramanujan disagreed, stating that it was actually quite interesting because it could be expressed as the sum of the cubes of two different numbers in two ways: 1^3 + 12^3 = 9^3 + 10^3 = 1729. Hence, 1729 holds historical significance as the smallest number with this unique property.