I would tell you, but this margin is too small

First notice that $125 = 5^3, 512 = 2^9 = 8^3, 1 = 1^3$ so the equation is $(5x)^3 = 1^3 + (8y)^3$ . Fermat's last theorem (the full result by Wiles is actually not necessary, as elementary proofs for the case $n=3$ were known by Euler) tell us no solutions exist for non-zero integers. So if a solution were to exist, $x=0$ or $y=0$ but then we would have, for example, $125x^3 = 1$ which clearly has no integer solutions.