<s>[INST] Cubes, squares and minimal sums 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!
 [/INST] 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.
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