This is a tough lesson to learn. Serial numbers are vital and need to be stored. If you purchased your programs directly from Adobe the serial numbers might be in your account history. Have you checked your Adobe account. If they aren't there I suppose you could contact Adobe through the chat system and see if there is something that can be done.
If you have LR5 on your old mac you should be able to get the info from the system files.
Launch Finder and click the Go menu whilst holding down the Opt(alt) key and choose Library
(n.b. the library is hidden until you press alt) then go to:
/Application Support/Adobe/Lightroom/Lightroom 5.0 Registration
Click choose application and select text editor then click open.