number theory - showing that the Diophantine $3x^2+2=y^2+6z^3$ equation has no solutions

I'd really love your help with showing that the Diophantine $3x^2+2=y^2+6z^3$ equation has no solutions.

I know that Diophantine equation of the form $ax+by+cz=d$ iff $\gcd(a,b,c) | d$, but how do I deal with the squares?

Any hints? suggestions?

Thanks!

Answer

If $(x,y,z)$ is a solution, then looking modulo $3$ you should have
$$y^2\equiv2\pmod3$$

It is then easy to see that there is no integer satisfying this equation.