If you took the time to register Lightroom 3 with Adobe you can log into your Adobe account and get the serial number from there. If you didn't register your product then you will have to call Adobe customer support.
Thanks Jim! Where do I find the contact information I've tried to connect with someone and all I get is links to many different categories. Do you have a contact phone number you can offer me?
If you still have access to the computer LR was installed on you can find both the LR3 & LR4 serial numbers here:
Registration data (serial number) is in //Library/Application Support/Adobe/Lightroom/Lightroom 4.0 Registration
Windows 7 Vista
Registration data (serial number) is located in the C:\ProgramData\Adobe\Lightroom\Lightroom 4.0 Registration.lrreg file.
Lightroom 4 reads the Lightroom 3.0, 2.0, or 1.0 registration registry entry when it validates upgrade serial numbers. If you install an upgrade version of Lightroom 4 and it does not find a file for either Lightroom 3.0, 2.0, or 1.0, it prompts you to enter the previous version serial number.
This are plain text files which can be opened in Window's Notepad or it's equivalent in OS X.
Thank you for your response. I'll see if I can open the old computer and find it