American Airlines is trying to boost its Facebook numbers and in doing so you could end up with some free frequent flier miles!

Facebook users who “like” the American Airlines AAdvantage Facebook page will earn between 100 miles and 100,000 miles as part of what’s being called the “Mystery Miles” promotion.

Simply input your AA frequent-flier number when the page pops up and the number of miles you’ll receive then appears. The promotion runs through March 31, 2011.

I’ve heard of at least one person who got 25,000 miles by participating. For the record, I got 100 miles. But hey, every little bit helps.

Here is the link: http://www.facebook.com/aadvantage