I was running Lightroom 3.6 with the OS Windows XP (Sp3). The whole system crashed, I am unable to recover the serial number. I then upgraded my OS to Windows 7 (x64) and down loaded Lightroom 4.4, I wanted to buy a licence so I then upgraded to Ligh

If you upgraded from 3.6 to 5 (you didn't indicate buying 4.4) then you will need the serial number from 3.6 in order to be able to use 5.  If you did not keep a record of it there is a chance it might be available thru your Adobe account online, but if not, there is likely not much you can do except to return the upgrade and purchase a full version.  Try contacting support using chat and explain your situation to see if they can help.

Serial number and activation chat support (non-CC)

http://helpx.adobe.com/x-productkb/global/service1.html ( http://adobe.ly/1aYjbSC )

If you did purchase 4.4 then that should be sufficient to use to activate 5.

I can never be sure what goes on with Lightroom, and the Adobe site (even when it works properly) makes finding information about products difficult these days so I am at a loss to confirm anything I say.