PAT25 would be a temporary military flight call sign, not a tail number. US military aircraft don't have FAA "N" registration numbers like civilian aircraft do. Military aircraft do have official numbers but those aren't used for ATC or radio communications.

In this particular case it appears that the "PAT" part is an acronym for Priority Air Travel. In other words, it was a VIP transport flight (there may not have been any VIPs actually on board at the time).