MINECRAFT will not let me play on two computers

I recently bought Minecraft on one of my computers, I downloaded the game and started playing. However, I have another computer and when I try to log in and play on the game, the game promptly "refreshes" itself and the game will not pop up. I checked and I was not logged in to both computers at the same time, it just won't work on this one computer. I have also updated the graphics driver, but this was not the problem either. It will let me sign in on the second computer, but when I hit the play button, the game acts like it will work but then it shows the play button again. The game also says its not crashing when it does this. I do not have any mods installed either.