you may have a corrupted "ProAppSystemID" file or Aperture cannot write to the System Library.
To stop Aperture asking for the serial number, first check, if you still have an old file "ProAppSystemID" in your System Library.
- Quit Aperture.
- Go to your MacintoshHD and open the folder "Library", then "Application Support", then "ProApps".
- Remove the File "ProAppsSystemID" to the Desktop, if it exist and restart Aperture.
- Reenter your serial number, hopefully for the last time.
Sometimes Aperture writes this file to a temporary directory, when it cannot write to the Library. If the above does not work, look at the temporary directory /tmp
Use the Finder's "Go" menu
Go > Got to folder and enter /tmp into the textfield. If you see the file there, move it to its proper location.
In this case you may want to check the permissions on the "ProApps" folder. You need Read and Write permissions. MAke sure you are using an administrator account, when entering the serial numbers.
I updated to Aperture 3.4.3 and still had the problem. Removing the file MacintoshHD > Library > Application Support > ProApps > ProAppsSystemID solved the problem.
The problem arose when my old Mac Pro died. I bought a new one and restored all apps and documents from Time Capsule. As noted, v.3.4.3 did not solve the problem.