Numbers that can be expressed as the sum of two cubes in two different ways are known as:
Hardy-Ramanujan numbers
The idea comes from a well-known story about the mathematician S. Ramanujan. When G. H. Hardy visited him and remarked that his taxi number, 1729, seemed dull, Ramanujan replied that it is the smallest number expressible as the sum of two cubes in two different ways.
Step 1 — \(1729 = 1^3 + 12^3 = 1 + 1728\).
Step 2 — \(1729 = 9^3 + 10^3 = 729 + 1000\). Other such numbers include 4104 and 13832; numbers of this kind are named after Hardy and Ramanujan (also called taxicab numbers).
Cube numbers are numbers of the form \(n^3\) such as 8 or 27; prime numbers have exactly two factors; Pythagorean numbers (triplets) satisfy \(a^2 + b^2 = c^2\). None of these describes a sum of two cubes in two ways.
Hardy-Ramanujan numbers option is correct.
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