Is it known that there exist no coprime positive integers $A$, $B$ and $C$ such that $A^3+B^4=C^3$? This is a particular case of [Beal's Conjecture](https://en.wikipedia.org/wiki/Beal%27s_conjecture).



Particular case of Beal's Conjecture
You can rewrite the diophantine equation in the form $C^3-A^3 = B^4,$ that is $C^3+ (-A)^3=B^4.$ 


Now, Bruin proved that integer solutions to $x^3+y^3 = z^n$ with $n \in \{4, \, 5\}$ and $xyz \neq 0$ satisfy $\textrm{gcd}(x, \, y, \, z) >1$, see 


N. Bruin, [On powers as sums of two cubes](http://link.springer.com/chapter/10.1007%2F10722028_9), Lecture Notes in Computer Science Volume **1838** (2000), pp 169-184. 


