Question

The number of positive integers with the property that they can be expressed as the sum of cubes of $2$ positive integers in two different way is?

• A. $1$
• B. $100$
• C. Infinite
• D. $0$

Answer 1

Answer: (C)

Infinite

Let $n$ be the positive integer with the property that it can be expressed as the sum of the cubes of $2$ positive integers in $2$ different ways. Then $n=a^3+b^3=c^3+d^3$.
Then any number of the form $m^{3}n$ also can be expressed as sum of the cubes of two integers in two different ways, as $m^{3}n={\left(ma\right)}^3+{\left(mb\right)}^3={\left(mc\right)}^3+{\left(md\right)}^3$. Since $m$ can take infinitely may values there will be $\infty$ such numbers available.
Now make sure that there exist at least one such number $n$.
Let, $n=1729=1728+1=1000+729$
Since there is at least one such $n$, there will be infinitely many such numbers.