Question

The smallest Hardy Ramanujan Number is:
(a) 1727
(b) 1728
(c) 1729
(d) 1730

Answer 1

Hint: Hardy Ramanujan numbers are the Numbers which are smallest positive integers which can be represented or expressed as sum of two positive numbers in n ways (at least 2 ways).

Complete step-by-step solution -
The Hardy Ramanujan number is a very fascinating number in mathematics. Hardy Ramanujan numbers are those numbers which are the smallest positive integers which can be represented or expressed as sum of 2 positive integers in n ways $\left( n>1 \right)$.
Now, Ramanujan’s n-way solution is a way in which a positive integer that can be expressed as sum of 2 cubes in n different way:-
He divided them in number of categories as
Ramanujan’s one way solution
Ramanujan’s two way solution
Ramanujan’s three way solution etc.

Ramanujan’s first way solution means that integers that are expressed as sum of 2 cubes ( in at least one way ). Some of these numbers are \[\left\{ 2,9,16,28,35,54\ldots \right\}\]
We can verify the given numbers in the above bracket as
$\begin{align}
  & {{1}^{3}}+{{1}^{3}}=2 \\
 & {{1}^{3}}+{{2}^{3}}=9 \\
 & {{2}^{3}}+{{2}^{3}}=16 \\
\end{align}$
Here, all these numbers can be expressed as a sum of 2 cubes in a single way.
Ramanujan’s second ways of solutions have integers that can be expressed as a sum of 2 cubes in more than 1 way (at least in 2 ways). Some of these numbers are $\left\{ 1729,4104,13832,20683\ldots \right\}$ verification can be done as
\[\begin{align}
  & 1729={{1}^{3}}+{{12}^{3}}={{10}^{3}}+{{9}^{3}} \\
 & 4104={{16}^{3}}+{{2}^{3}}={{15}^{3}}+{{9}^{3}} \\
 & 13832={{24}^{3}}+{{2}^{3}}={{20}^{3}}{{18}^{3}} \\
\end{align}\]
Here all these numbers can be expressed as a sum of 2 cubes in 2 different ways.
Ramanujan’s three way Solution:
Integers that can be expressed as sum of 2 cubes in at least 3 ways, Numbers are given as $\left\{ 87539319,119824488,143604279,\ldots \right\}$
Verification can be done as $87539319={{167}^{3}}+{{436}^{3}}={{228}^{3}}+{{423}^{3}}={{255}^{3}}+{{414}^{3}}$
Similarly we can get some numbers with a 4 way solution as well. So, as per the definition of Hardy Ramanujan’s number, we get that number.
1729 is the smallest integer which can be represented in the form of two cubes in two ways.
So, option (A) is correct.

Note: Ramanujan number’s one solution does not come under the category of Hardy Ramanujan number. It contains only the numbers which can be expressed in two numbers in n ways with $n>1$.