Solve the equation $19 x^{3}-84 y^{2}=1984$ in integers.

#
The equation can be rewritten as $19\left(x^{3}-100\right)=84\left(1+y^{2}\right)$. The right-hand side is divisible by 7, so the left-hand side must also be divisible by 7. Therefore,

$x^{3}-100$ is divisible by 7, which means $x^{3} \equiv 2(\bmod 7)$. However, cubes modulo 7 yield only remainders of 0 or $\pm 1$.

## Answer

No solutions.