Probability of coprimality / GP random()

[hermann@stamm-wilbrandt.de]

Last week I visited Heidelberg University library.
I found area where books on Algebra, analytic number theory,
p-adic numbers, ... were located and picked 5 books.
Looking through I learned about subject in one of the books.
Now I found it on Wikipedia for reference:
https://en.wikipedia.org/wiki/Coprime_integers#Probability_of_coprimality

The probability is 6/Pi^2 (roughly 61%), and I tested GP.
I am pretty sure that GP random() passes other primality tests as well.
Nice to see that the results below seem to be as they should.


The result of 6/Pi^2 is for computing the Basel problem.
Today I found a nice geometrical "proof" for 6/Pi^2:
https://www.youtube.com/watch?v=d-o3eB9sfls&t=635s

And thought I might post here.

Regards,

Hermann.

hermann@7950x:~$ gp -q
? #
   timer = 1 (on)
? 
N=50000;M=10^100;c=0;for(i=1,N,if(gcd(random(M),random(M))==1,c+=1));print(c*1.0/N);print(6/Pi^2);
0.61004000000000000000000000000000000000
0.60792710185402662866327677925836583343
time = 65 ms.
? 
N=500000;M=10^100;c=0;for(i=1,N,if(gcd(random(M),random(M))==1,c+=1));print(c*1.0/N);print(6/Pi^2);
0.60770200000000000000000000000000000000
0.60792710185402662866327677925836583343
time = 647 ms.
? 
N=5000000;M=10^100;c=0;for(i=1,N,if(gcd(random(M),random(M))==1,c+=1));print(c*1.0/N);print(6/Pi^2);
0.60759480000000000000000000000000000000
0.60792710185402662866327677925836583343
time = 5,729 ms.
? 
N=50000000;M=10^100;c=0;for(i=1,N,if(gcd(random(M),random(M))==1,c+=1));print(c*1.0/N);print(6/Pi^2);
0.60792814000000000000000000000000000000
0.60792710185402662866327677925836583343
time = 58,623 ms.
?