I'll be grateful for any help with the foollowing question. I think the solution must be easy enough but i haven't figured it out yet.
Let a and b be positive integers such that
1) $\exists c \in \mathbb{Z}: ~~ a^2 + b^2 = c^3$
2) $\exists d \in \mathbb{Z}: ~~ a^3 + b^3 = d^2$
The task is to find the maximal possible sum of a and b.
Thanks in advance.
Suppose that we found such integers $a$ and $b$. Let $a_1=k^6a$, and let $b_1=k^6b$.
Then $a_1^2+b_1^2=(k^4c)^3$ and $a_1^3+b_1^3=(k^9 d)^2$.
Now note that $a=2$, $b=2$ have the property that $a^2+b^2$ is a perfect cube and $a^3+b^3$ is a perfect square.
So already by taking $k=1,2,3,\dots$, we have infinitely many examples. There is no maximum value for the sum $a+b$.
Remark: Perhaps by putting additional constraints on $a$ and $b$, we can come up with a natural problem of the same kind for which there are only finitely many solutions.