<s>[INST] When is the sum of two squares the sum of two cubes When does $a^2+b^2 = c^3 +d^3$ for all integer values $(a, b, c, d) \ge 0$. I believe this only happens when: $a^2 = c^3 = e^6$ and $b^2 = d^3 = f^6$. With the following exception:


*

*$1^3+2^3 = 3^2 + 0^2$


Would that statement be correct?
Is there a general formula for when this happens?
 [/INST] *

*$a=x^3-3x^2y-3xy^2+y^3,b=x^3+3x^2y-3xy^2-y^3,c=d=x^2+y^2.$


*$a=3(x^3-3xy^2),b=3(3x^2y-y^3),c=x^2+y^2,d=2(x^2+y^2)$
$\cdots$
See this post.
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