Cubes, squares and minimal sums

Question

I have trouble solving the following task: i need to find positive integers a and b such that

1) $a \neq b$

2) $ \exists c \in \mathbb{N} : ~ a^2 + b^2 = c^3$

3) $\exists d \in \mathbb{N}: ~ a^3 + b^3 = d^2$

4) sum $a + b$ is minimal possible

Thanks in advance!

Answer 1

If $a=1250$ and $b=625$, then $a=2\cdot 5^4$ and $b=5^4$, and

$$ a^2 + b^2 = 2^2\cdot 5^8 + 5^8 = (4 + 1)5^8 = 5^9 = (5^3)^3 $$ $$ a^3 + b^3 = 2^3\cdot 5^{12} + 5^{12} = (8 + 1)5^{12} = 3^2(5^6)^2 = (3\cdot 5^6)^2 $$

This was the smallest example I found by an extremely primitive computer search.