<s>[INST] Positive integer solutions to $a^3 + b^3 = c^4$ Let $n$ and $m$ be positive integers. We know that
$n^3 + m^3 = n^3 + m^3$
Multiply both sides by $(n^3 + m^3)^3$; on the LHS you distribute, and on the RHS you use power addition rule;
$(n^4 + nm^3)^3 + (m^4 + mn^3)^3 = (n^3 + m^3)^4$
So we get an infinite array of solutions with
$a = n^4 + nm^3$
$b = m^4 + mn^3$
$c = n^3 + m^3$
Are there any solutions to the equation (in the title) that cannot be expressed in this form?
 [/INST] $$793^3+854^3=183^4$$ and $$183$$ is not the sum of two cubes.
</s>