TITLE: Diophantine equation not solvable in $\mathbb{Q}$, but in $\mathcal{O}_p$
QUESTION [5 upvotes]: I'm trying to think of an example of a diophantine equation which can be solved in $ \mathcal{O}_p$ (meaning it can be solved $\mod p^k$ for all $ k $) for all prime $ p $'s, but not in $\mathbb{Q}$ 
I don't really think that's such an easy task - and probably there is some classic answer, but I can't think of one.
I would appreciate some help

REPLY [4 votes]: Here's one I read a number of years ago
that works mod $p$ for every prime $p$.
I don't know if it works for $\mathcal{O}_p$.
Let
$f(x)
=(x^2-2)(x^2-3)(x^2-6)
$.
$f$ obviously has no rational roots.
If
$x^2-2$ and $x^2-3$
have no roots mod $p$,
then 2 and 3 are not
quadratic residues mod $p$,
so 6 is a quadratic residue mod $p$,
so $x^2-6$ has a root.