What happened before Aperture started to ask for the serial number? Did you change the hardare in any way? Reinstall the system?
If the problem is accidental and your hardware and software did not change at all, you should be able to restore the licence by restoring the System Library and your User library from a backup of the time, when Aperture has been working properly. If that succeeds, write the licence number down. It is displayed in the "About" panel: From the main menu bar: Aperture > About Aperture.
If you cannot restore the library files from your backup, contact Apple with a proof of purchase fr a replacement serial number:
What is your current Aperture version btw? Are you asking about upgrading from Aperture 2 or an earlier version of Aperture 3?
Thank you for your helpful response. In answer to your question, I didn't alter my system or perform a reinstall. I do have a Time capsule back up but I've chosen the Pro Application Replacement Serial Number route since I have a copy of my original receipt. I have an earlier version of Aperture 1.x, which I purchased in 2010. Once I get Aperture back on-line, I intend to upgrade. Thanks again, Leonie.