Explanation:

The Hardy-Ramanujan number, also known as the Taxicab number, is a number that can be expressed as the sum of two positive cubes in two different ways. It is named after the British mathematician G.H. Hardy and the Indian mathematician Srinivasa Ramanujan, who studied these numbers.

In order to determine which of the given numbers is a Hardy-Ramanujan number, we need to check if the number can be expressed as the sum of two positive cubes in two different ways.

Checking the options:

a) 1724:

To check if 1724 is a Hardy-Ramanujan number, we need to find two pairs of positive integers (a, b) and (c, d) such that a^3 + b^3 = c^3 + d^3 = 1724.

However, after trying different combinations, we cannot find such pairs. Therefore, 1724 is not a Hardy-Ramanujan number.

b) 1725:

Similar to the previous option, we need to find two pairs of positive integers (a, b) and (c, d) such that a^3 + b^3 = c^3 + d^3 = 1725.

Again, after trying different combinations, we cannot find such pairs. Therefore, 1725 is not a Hardy-Ramanujan number.

c) 1727:

Once again, we need to find two pairs of positive integers (a, b) and (c, d) such that a^3 + b^3 = c^3 + d^3 = 1727.

After trying different combinations, we cannot find such pairs. Therefore, 1727 is not a Hardy-Ramanujan number.

d) 1729:

To determine if 1729 is a Hardy-Ramanujan number, we need to find two pairs of positive integers (a, b) and (c, d) such that a^3 + b^3 = c^3 + d^3 = 1729.

Interestingly, we can find two pairs that satisfy this condition:
1^3 + 12^3 = 9^3 + 10^3 = 1729

Therefore, 1729 is a Hardy-Ramanujan number.

Conclusion:

Out of the given options, only 1729 is a Hardy-Ramanujan number. The other options (1724, 1725, and 1727) do not satisfy the condition of being expressible as the sum of two positive cubes in two different ways.