# Is there a solution to the diophantine equation $a^3 + b^3 = c^2$ other than $1^3 + 2^3 = 3^2$ or scalings thereof? [duplicate]

My question concerns the diophantine equation $a^3 + b^3 = c^2$. I know one solution: $1^3 + 2^3 = 3^2$. But this is special in (at least) two ways: the $a$ and $b$ are not coprime; the solution is a special case of the identity: sum of the first $n$ cubes $=$ the square of the $n$th triangular number.

Are there other solutions?

If not, is there an elementary proof of the fact?

## marked as duplicate by Namaste, Adrian Keister, Cesareo, Nosrati, BrahadeeshAug 8 '18 at 13:41

For example, $(2, 2, 4), (7, 21, 98), (65, 56, 671),$ etc.