Here it is (448 bytes):

int m=1811939329,N=1,t[1<<26]={2},a,*p,i,e=73421233,s,c,U=1;g(d,h){for(i=s;i<1<< 25;i*=2)d=d*1LL*d%m;for(p=t;p<t+N;p+=s)for(i=s,c=1;i;i--)a=p[s]*(h?c:1LL)%m,p[s] =(m*1U+*p-a)*(h?1LL:c)%m,*p=(a*1U+*p)%m,p++,c=c*1LL*d%m;}main(){while(e/=2){N*=2 ;U=U*1LL*(m+1)/2%m;for(s=N;s/=2;)g(136,0);for(p=t;p<t+N;p++)*p=*p*1LL**p%m*U%m; for(s=1;s<N;s*=2)g(839354248,1);for(a=0,p=t;p<t+N;)a+=*p<<(e&1),*p++=a%10,a/=10; }while(!*--p);for(t[0]--;p>=t;)putchar(48+*p--);}

This program computes 2^{74207281}-1, which is the biggest known
prime number (about 23 million digits !). For more information about
how it was found and who found it, look at the GIMPS Project .

I compiled it successfully with gcc with Linux. In order to
compile it, your C compiler must support the 64 bit *long long*
type.

In order to have a small and yet asymptotically efficient code, I
decided to do the computation of 2^{N}-1 directly in base
10. The power involves repeated squarings and multiplications by
two. The squarings are implemented by doing fast convolutions using
a Number
Theoretic Transform.

A previous version of this program to compute 2^{6972593}-1
won the International Obfuscated C Code
Contest of Year 2000.

Thanks to Marco Cecchi who suggested some syntactic changes to save a few characters.

This program is Freeware.

Fabrice Bellard - http://bellard.org/