>

restart;

>	second:=evalf(5^(3125)); # This we can calculate...

$\mathrm{second}:=.1911012598e2185$

>	mult:=evalf(%/3125); # Next, we can take that huge number and divide it to # parts by 3125 (that we can calculate). # That way we get how many numbers 1.911e2184 needs to be # multiplied by theirself.

$\mathrm{mult}:=.6115240314e2181$

>	n1:=6.115*1.911*10^2180; n2:=2184*2180;

$\mathrm{n1}:=.1168576500e2182$

$\mathrm{n2}:=4761120$

>	third:=n1*10^n2;

$\mathrm{third}:=.1168576500e4763302$

>	# We have an aproximation of 3rd step. Now we can say # something about a power of 5^(5^(5^(5^5))) = 5^third # Like before, we can take 'third' number and divide it # into parts that we know, what power they are. mult2:=third/3125;

$\mathrm{mult2}:=.3739444800e4763298$

>	f1:=1.911*3.739445*10^2180; f2:=2184*4763297;

$\mathrm{f1}:=.7146079395e2181$

$\mathrm{f2}:=10403040648$

>	fourth:=f1*10^f2;

Error, numeric exception: overflow

>	10403040648+2180;

$10403042828$

>	# It's huge. Number itself would fit in 10.5GB of diskspace # if my thinking was good. PLEASE check it and write if any mistake # (in method) occured. # If You're interresed in large prime numbers, the biggest I know # is in power of 10^24... 2^30*3^30+7; isprime(%);

$221073919720733357899783$

$\mathrm{true}$

>	# It's PROPABLY prime, test is propabilistic...