For every positive integer $n$, is there some number field $K$ and elliptic curve $E/K$ such that $E(K)$ has rank $n$?

It's easy to show that the set of such $n$ is unbounded. But can one show that *every* positive integer is the rank of some elliptic curve over a number field?

The analogous question for a fixed number field is expected to have a negative answer (c.f. e.g., this question) but is still conjectural. But I wonder if one might be able to prove a positive answer to the question I asked above.

fixednumber field is similarly useless, and yet it's a very interesting open problem. $\endgroup$ – Wojowu Jun 26 '19 at 15:22