Baltic Way 2017 · Problem 17

We consider the equation modulo 13 since both 3 and 4 divides $12=13-1$. Now $x^{4} \bmod 13 \in\{0,1,3,9\}$ and $y^{3} \bmod 13 \in$ $\{0,1,5,8,12\}$. We can verify that $x^{4}+y^{3} \not \equiv 7 \bmod 13$. However $z !+7 \equiv 7 \bmod 13$ if $z \geq 13$, what leads to a conclusion that this equation has no solutions with $z \geq 13$, what proves that it has finite number of solutions.