Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.)
We are very far from proving the existence of such an integer, let alone find an explicit value which works.
My question is:
What is the best known lower bound for $n$?
One way to obtain a lower bound $m$ for $n$ is to prove the existence of a curve of genus $2$ over $\mathbf{Q}$ with at least $m$ rational points.
Best Answer
I believe it is 642. See http://www.mathe2.uni-bayreuth.de/stoll/recordcurve.html