Can my Home PC Handle the Lucas-Lehmer Test by Itself? (or Do I Need GIMPS)
LLT implementation in Pari/gp
p = 19937; Mp = 2^p-1; x = 4;
moduloM(p,n) = { a= shift(n,-p); b = n-shift(a,p); r = b+a; };
for(n=1, p-2, x = moduloM(p,moduloM(p,x^2-2)));
/* slow version :
x = Mod(4,Mp); for(n=1, p-2, x = x^2-2); */
if(x == Mp, print("2^",p,"-1 is prime"), print("2^",p,"-1 is NOT PRIME"));
$25$ million digits is large but within reach of some number theory tools like pari/gp or maple.
So, in principle, you can check such a number with a single normal computer.
Of course, it will take long until the number is checked.
Note that the calculations that have to be done do not exceed the $50$ million digits mark.