Question

Which of these is the smallest Hardy-Ramanujan number?

• A. 730
• B. 1728
• C. 1729
• D. 4104

Answer 1

The correct option is C 1729

Hardy-Ramanujan numbers are the numbers that can be expressed as the sum of two cubes in two different ways.
$730=9^3+1^3$
$1728={12}^3$
$1729=9^3+{10}^3={12}^3+1^3$
$4104={16}^3+2^3={15}^3+9^3$
So, of the given numbers 1729 and 4104 are Hardy-Ramanujan numbers.
And the smallest Hardy-Ramanujan number is 1729.

So, the option (3) is correct.