Diophantine equation not solvable in $\mathbb{Q}$, but in $\mathcal{O}_p$

Question

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

Answer 1

In this paper is mentioned $3x^3+4y^3+5z^3=0$, referred to there as 'Selmer's example' http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/selmerexample.pdf

Answer 2

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.