How to prove that $x^2 - 3y^2 = 11$ is not possible as long as x and y are integers.

Question

Really No idea how I can go about solving it. What I did was,

$$y = \sqrt{\frac{x^2 - 11}{3}}$$

but cant go beyond that. Can anybody help?

Answer 1

$$x^2 \equiv 0, 1 \pmod 3$$

in fact $0^2 \equiv 0, 1^2 \equiv 1, 2^2 = 4 \equiv 1 \pmod 3$

of course $3y^2 \equiv 0 \pmod 3$ (it is divisible by $3$)

so

$$x^2 - 3y^2 \equiv 0, 1 \pmod 3$$ but $$11 \equiv 2 \pmod 3$$ so it is impossible.