American Airlines has started posting deals to it’s Facebook and Twitter pages. Details of the Facebook promo are below.

To celebrate the recent launch of social media channels dedicated to earning miles in the AAdvantage® program, American Airlines is launching the AAdvantage “Mystery Miles” promotion. Through March 31, 2011, members can simply “like” the AAdvantage program on Facebook at facebook.com/AAdvantage and quickly earn between 100 and 100,000 miles. The actual number of miles members will earn is the mystery. Miles will be deposited in the memeber’s AAdvantage account in 3-5 business days.

Link to Promo on Facebook.