What qualities must a mineral possess in order to be considered a gem?

It depends on 4 characteristics:
Beauty
Durability
Hardness
Rarity.
Beauty is the clearness (clear doesn't always mean worth more), color and brilliance, aka the brightness. Durability and hardness are quite the same, the harder and more durable, the better (think of diamonds are the hardest known natural material). Then rarity. Some minerals are very rare (big diamonds). The rarer, the pricier of course. Also the way it is cut makes a big difference.

668900 atoms in total

The mass has escaped into a gas which you can not see so the law stays true.