Your browser is doing the downloading. The URLs are available from the API (I changed random characters):

  $ curl -I https://graph.facebook.com/100000624321700/picture
  HTTP/1.1 302 Found

(...)

  Location: https://fbcdn-profile-a.akamaihd.net/hprofile-ak-prn1/573867_100000621711234_930571234_q.jpg

Then it is just a matter of incrementing the number. This might be a fun research project.