If you bought Aperture 3 recently from the App Store, there is no serial number and Aperture should simply run without requiring a serial number. Why are you asking? Does Aperture ask for a serial number, when you launch it?
Yes, I've got the same problem, Aperture 3 from the App-store. Was working fine under 3.4.2 then they go and update to 3.4.3 which according to the change-log only is supposed to be fixing this problem for 3.4.2!!!