Character mistake: At the end of the movie, the rescue plane changes its callsign to Air Force One because the protocol is "Air Force One" is the callsign for any U.S. military plane (or, more specifically, an Air Force Plane) with the President on board, not just the blue and white 747. Thus, it would not be used for a plane on which the president is not on board. This is an important detail considering that for most of the middle section of the film, the terrorists believe the president has escaped and that they are dealing with a random secret service agent resisting them. The vice president and other administration officials dealing with the terrorists don't want them to know the president is still on board, as it could motivate the terrorists to threaten his family further. So, when the fuel tanker shows up to refuel the plane and addresses it as "Air Force One" to give instructions on the procedure, they are inadvertently confirming that the president is on board. To maintain the ruse, they should use the callsign "SAM-28000" or "Air Force 28000" when talking to the terrorists, referring to the plane's tail number. Similarly, any time an official makes a statement about the incident in public, they could refer to the plane as "28000" to keep up the ruse to the press (though it's not uncommon to refer to the 747s as "Air Force One" for the sake of simplicity in casual or non-official capacities, an instance of one plane communicating with another would not be).
Plot hole: At the start when the guys are infiltrating that building they parachute down to the roof. They are dressed in black so they can't be seen, but their parachutes are white. (00:02:50)
Trivia: One of the secretaries tells the president she can send a fax to the White House, and he replies something like: "If this works, I'll make you Postmaster General". In the closing credits, she's credited as "Future Postmaster General".
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