Is your problem with the version 6 serial number, or when you are asked for the version 5 serial number to prove that you qualify to install and upgrade version?
-and more help in http://helpx.adobe.com/creative-suite/kb/invalid-serial-number-cs4.html
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Did you purchase a Lightroom 6 upgrade, or are you trying to use your Lightroom 5 serial number? Lightroom 6 is an upgrade that must be purchased. If you already know that, I apologize.
I bought at version 5 a couple years ago. I have updated a few times in the "5" world but don't remember needing my serial number (but that was a while ago and may have without issue). So my serial number is of the "5" vintage. I tried the solution #4 earlier, and it was starting to work. I successfully installed the Trial, but when I went to open Lightroom 6 and chose my latest catalog, it didn't load the photos. The dialog box in the upper right said "calculating" but it didn't accomplish anything for several minutes. It could find the photo files in Finder, so I know they were associated, but they never opened (5 minutes maybe). I got nervous and uninstalled it for now, sticking with 5.7.1 until I get better info.
Ah! I did not know that, but that is probably the answer! Thanks.