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How do I do very big integer computing by Mathematica?

• To: mathgroup at smc.vnet.net
• Subject: [mg115335] How do I do very big integer computing by Mathematica?
• From: a boy <avvboy at gmail.com>
• Date: Sat, 8 Jan 2011 03:37:07 -0500 (EST)

I'm going to search big Fibonacci prime numbers. I think there is a
simple primality test algorithm for Fibonacci number, like  Lucas=96
Lehmer primality test for 2^n-1 . I'm lazy and don't want to write
many codes. So i want to ask:

p=43,112,609;
s[0]=4;
s[n_]:=s[n-1]^2-4
pt=Mod[s[p-2], 2^p-1]==0

how do I compute s[43,112,609-2] directly? It seems the largest
integer in M~ is 2^32^32, isn't it?