Hello =) At http://math.stackexchange.com/questions/1122481/the-diophantine-equation-y2-x37-has-no-solutions/1122506#1122506 in the first step, to show that $x$ cannot be even, can we also do it using $\pmod 4$ ???
$$x=2k \Rightarrow y^2=8k^3+7 \equiv 3 \pmod4$$
Since
$$y^2 \equiv 0,1 \pmod4 \equiv 3 \pmod4$$
$x$ cannot be even. Is this also correct???