If you have a subscription, Lightroom does not need your serial number - it needs for you to just log in with your Adobe ID. If it is asking for your serial number, you have the wrong version of Lightroom installed. You need to uninstall Lightroom and then re-install it from the Adobe Creative Cloud App instead.
If Lightroom is prompting you for a serial number then you have installed your old standalone copy of Lightroom. The creative cloud subscription plan relies on your Adobe ID and password. You must install both Lightroom and Photoshop from the creative cloud application manager.
I think we posted at the same time, Rick. I wasn't trying to upstage you.