Q: if the box is lost, how can I get my serial number?

You mean the Aperture serial number?

If you have a second installation of Aperture, you can see the serial number in the "About" panel from the main menu bar: Aperture > About Aperture.

Else you will have to contact Apple with a proof of purchase for a replacement serial number, see the following support page:

Pro Application Replacement Serial Numbers

Regards

Léonie

Do you have a time machine backup? If so you can restore it from there.

Do you have a time machine backup? If so you can restore it from there.

That will only work, if you are not reinstalling on a new system. The "ProAppSystemID" file ties the registration number to the current machine.

Great! May I suggest to write a copy of the numbers and  keep them in a safe place? Just in case ....

That is the reason I keep ALL my installed software copied on a disk - including the serial numbers in a text file.

This way it is easy to find the media, serial number and the installation is pretty fast.

Lately I have been using 32GB USB sticks.