No integer solution questions?

Question

1. Prove that $x^2-7y^2-24=0$ has no integer solutions.
2. Prove that $x^2+(x+1)^2+(x+2)^2=y^2$ has no integer solutions.

I am unsure of how to approach these problems? Are they modulo questions? Do I factor?

Answer 1

2) $$y^2=3x^2+6x+5,$$ which says $$y^2\equiv2(\mod3),$$ which is impossible.

1) If so we have $$x^2\equiv3(\mod7).$$

Can you kill it now?