<s>[INST] Show that there are no rational solutions on the ellipse $2x^2 + 3y^2 = 1$ Show that there are no rational solutions on the ellipse $2x^2 + 3y^2 = 1$.
Want to make sure that my proof is correct.
Suppose there are rational solutions.
Note $2x^2 + 3y^2 = 1 \iff 2x^2 + 3y^2 = z^2\ |\ x,y.z \in \mathbb{Z}$
Since taking $(\frac{x}{z}, \frac{y}{z})$ as our rational solution, $2(\frac{x}{z})^2 + 3(\frac{y}{z})^2 = 1 \implies 2x^2 + 3y^2 = z^2$.
Now reducing $mod\ 3$, we have $2x^2 + 3y^2 \equiv  1 \ mod\ 3 \implies 2x^2 \equiv 1 \ mod\ 3 \implies -1x^2 \equiv 1\ mod\ 3 \implies x^2 \equiv -1\ mod\ 3 $, which is impossible.
 [/INST] As José Carlos Santos was pointing out in the comments, we have 
$$2x^2 \equiv z^2 \mod 3$$
But this does not imply your statement that 
$$2x^2 \equiv 1 \mod 3$$
If $z$ were divisible by $3$ and we would have 
$$2x^2 \equiv 0 \mod 3$$
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