Sum of 2 Squares in 2 Distinct Ways which is also Sum of Cubes

Theorem
The smallest positive integer which is both the sum of $2$ square numbers in two distinct ways and also the sum of $2$ cube numbers is $65$:

Proof
From Sum of 2 Squares in 2 Distinct Ways, the smallest $2$ positive integer which are the sum of $2$ square numbers in two distinct ways are $50$ and $65$.

But $50$ cannot be expressed as the sum of $2$ cube numbers:

Hence $65$ is that smallest number.