You should NEVER give any serial number away as it is pure money.
No one is entitled to require private information.
Look also to the Yellow Stripe on the opening page here:
We have received a number of reports where some members of this community are posing as Adobe Employees by offering to help troubleshoot issues personally using technologies such as Skype as a means to gain access to your computer and potentially your personal information. All Adobe employees are required to have an orange ‘STAFF’ badge next to their user profile which can be seen if you mouse-over their profile picture.