some other solutions:

• Fibonacci numbers (again)
• Fibonacci Numbers

On a related note, I have a pdf titled "Perlgolf History" (first page is a big image of a squaking ostrich if anyone knows of it) -- i don't remember where I d/l'd it from and at the moment w/just a brief google search I can't find it..

But anyways, in its "Terje's PGAS season 0" section, it covers "Modular Fibonacci" (note that 2m is '2<super>m</super>'):
5.12. Modular Fibonacci

http://terje2.perlgolf.org/~pgas/score.pl?func=rules&hole=12&season=0
The game started 2002-10-21 21:30:00 and ended 2002-10-27 19:00:00.
5.12.1. Rules
The Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...) are defi
+ned by the recurrence:
F1 = 1
F2 = 1
Fi = Fi-1 + Fi-2 for i > 2
Write a program which calculates Mn = Fn mod 2m for given pair of n an
+d m. 0 < n < 10000 and 0 <
m < 20 . Note that a mod b gives the remainder when a is divided by b.
Input consists of one newline-terminated line specifying a pair of n a
+nd m separated by a space.
Output should be corresponding Mn, and a newline.
Sample Input
11 7
Sample Output
89
[download]

5.12.2. Solutions