Every starting number ever tested by computer, into the astronomically large, reaches one, but no proof rules out either an unbounded starting number whose path never terminates or a hidden cycle other than the trivial one at 1-4-2; Paul Erdos himself is widely quoted as doubting mathematics currently has the tools to settle it.
What would resolve this A general proof covering every positive starting integer, or the discovery of a starting number that provably never reaches one (an unbounded trajectory or a second cycle).
OpenNumber theory / dynamical systemsWolfram MathWorld