Question

Find the natural number with the property that it can be expressed as the sum of the cubes of two natural numbers in two different ways.

Answer 1

There is no other way of writing them.

The number that can be expressed in sum of two cubes in 2 different ways is given by trial and error method :

$1^3+2^3=9$

$1^3+3^3=28$

$1^3+4^3=65$

$1^3+5^3=126$

$1^3+6^3=217$

$1^3+7^3=344$

$1^3+8^3=513$

$1^3+9^3=730$

$1^3+{10}^3=1001$

$1^3+{11}^3=1332$

$1^3+{12}^3=1729$ This can be represented as $9^3+{10}^3=1729$

So it is $1729$