Do you still have Lightroom 4 installed on the old computer? With Windows Explorer, go to C:\ProgramData\Adobe\Lightroom.
In this folder there will be a file named Lightroom 4.0 Registration.lrreg
Open this file in Notepad, and you should find the serial number. You might be able to find this file even if LR4 has been uninstalled.
The other option would be to log in to your Adobe account.
If you registered LR4, you can find the serial nunber there.
See the link below for help in locating your serial number. If you registered your product the serial numbers can be recovered from your Adobe account.
You should also be able to retrieve the numbers from the OS registry on your Laptop, do a search on the internet.
The laptop is a MacBook, any idea where I'd be able to find a similar file? Away from the computer at the moment so I can't attempt a fix until later.
Never registered the product so that's not an option unfortunately. Needless to say I'll never not register ever again.
I assumed you were on Windows.
For a Mac, try this folder: Library/Application Support/Adobe/Lightroom
There it is. I was sure had to be buried somewhere. Many thanks all!