The Collatz Conjecture has some money assigned to it.
Define $T(n) = \begin{cases} n/2 & n=0 \mod 2 \\ 3n+1 & n=1\mod 2 \end{cases}$
For $n\in \mathbb{N},$ does there exist a $N,$ dependent on $n,$ so that for every $T^N(n)=1?$
Lothar Collatz Conjectured the question in 1937 and it has been open since. Paul Erdos believing this to be a tough problem offered $$500 for it.
Edit: Fixed some dumb errors in the answer. First, Paul Erdos and not Paul Halmos offered the $500 as per wikipedia and several other informal references. Second, there was an elementary logic error in the statement.