Hi Camilla ...
If you downloaded and installed the trial version prior to downloading Aperture from the App Store, you need to delete the trial version.
It takes more then just dragging the app to the Trash. You need to use a utility that will delete the app and all the associated files > Download AppCleaner for Mac - Uninstall your apps easily. MacUpdate.com
Read here > Apple - Aperture - Trial "If you currently have a copy of the Aperture 3 Trial installed on your Mac, you must delete it from your Applications folder before downloading Aperture 3 from the Mac App Store."
After you delete the trial version you shouldn't be asked for a license or serial number.
That is not the issue..
I tried to find a billing to check for a serial number, and then I saw that the computer here is preinstalled with Aperture 3.. so what to do.. ? I don't recall getting any other serial numbers but the one on the billing for the computer ... And that's not valid?... any suggestions ?