Question

Which of the following is a Hardy Ramanujan number?
A) 1728
B) 4194
C) 13832
D) 25683

Answer 1

Hint: A Hardy Ramanujan number is a popular number which can be expressed as a sum of two cubes in two ways. So, we can test each number based on this definition to find the answer.

Complete step-by-step answer:
As we know that, any number x qualifies as a Hardy Ramanujan number, if
$x = {a^3} + {b^3} = {c^3} + {d^3}$
Where a, b , c and d are 4 different positive numbers. This combination is very rare.
If we try one by one option given, then for 13832, we can find two such pairs of numbers.
First pair is 2 and 24, because
$
  13832 = {2^3} + {24^3} \\
   \Rightarrow 13832 = 8 + 13824 \\
   \Rightarrow 13832 = 13832 \\
 $
Then second pair is 18 and 20, because
$
  13832 = {18^3} + {20^3} \\
   \Rightarrow 13832 = 5832 + 8000 \\
   \Rightarrow 13832 = 13832 \\
 $
It is clear that 13832 is the given Hardy Ramanujan number. Thus option C is correct.

Additional Information: This number derives its name from an interesting story. Mathematician G. H. Hardy once told about the great Indian mathematician Ramanujan. Once, while travelling in the taxi from London, Hardy noticed its number which was 1729. He had thought about it very little. When he entered the room where Ramanujan lay in bed. Hardy scarcely said hello to him. He was disappointed and declared that number 1729 as a dull number. Also, he added that he hoped that wasn't a bad omen. But, Ramanujan said that it is a very interesting number. Because it is the smallest number expressible as the sum of two positive cubes in two different ways.

Note: The smallest such number is 1729. As we can represent it in two different ways as a sum of two cubes. It is given by: $1729 = {1^3} + {12^3} = {9^3} + {10^3}$