Show that $y^2=x^3+1$ has infinitely many solutions over $\mathbb Z_p$. I first compared it with how I would solve this over the real numbers. You would  say:


*

*$y^2=\alpha$ has a solution for all $\alpha>0$, of which there are infinitely many. 

*$x^3+1>0$ for all $x>-1$, of which there are also infinitely many.


However I can't seem to extend this way of thinking to $\mathbb Z_p$. I have a strong hunch that I need to use Hensel's Lemma in some way, but I just can't see how. 

You want to show that the square root $\sqrt{x^3 + 1}$ exists in $\mathbb{Z}_p$ for infinitely many $x$. To do this, show that if $p \mid x$, then
$$\sqrt{1 + x^3} = \sum_{n \ge 0} {1/2 \choose n} x^{3n}$$
converges in $\mathbb{Z}_p$. (The case $p = 2$ is, as usual, a bit different, but not so bad.) 
