Re: PARI/GP timings for operations on biggest known 41,024,320 decimal digit prime

[hermann@stamm-wilbrandt.de]

On 2024-11-17 14:54, hermann@stamm-wilbrandt.de wrote:

Regarding the 3-line GP script at end of my previous email:
It got split onto 57 lines by mailing list software.

But Linux tr is GP's friend, split script can be executed directly with 
gp:

hermann@7950x:~/llr405src/linux64llr$ wc --lines Q.gp
57 Q.gp
hermann@7950x:~/llr405src/linux64llr$ gp -q < <(tr -d '\n' < q.gp)
1
hermann@7950x:~/llr405src/linux64llr$


> On 2024-11-17 03:08, Kurt Foster wrote:
> > In any case, the prime M splits into two prime ideals in Q(sqrt(-23)).
> >  If M = x^2 + 23*y^2 the ideals are principal; otherwise not.
In addition to the questions from my previous email,
are you sure that this applies to Mersenne primes as well?

I tested the Mersenne exponents with -23 being quadratic
residue up to 607 manually in gp, see below.

And for M_18..M_38 with -23 being quadratic residue with automation, 
below.

For all 18 primes sqrt(Mod(-23,p)) got computed and verified, not a 
single exception ...


Regards,

Hermann.

? p=2^607-1;
? s=lift(sqrt(Mod(-23,p)));
? (Mod(23,p)^((p+1)/4))^2==Mod(-23,p)
%22 = 1
? p=2^107-1;
? s=lift(sqrt(Mod(-23,p)));
? (Mod(23,p)^((p+1)/4))^2==Mod(-23,p)
%25 = 1
? p=2^89-1;
? s=lift(sqrt(Mod(-23,p)));
? (Mod(23,p)^((p+1)/4))^2==Mod(-23,p)
%28 = 1
? p=2^19-1;
? s=lift(sqrt(Mod(-23,p)));
? (Mod(23,p)^((p+1)/4))^2==Mod(-23,p)
%31 = 1
? p=2^13-1;
? s=lift(sqrt(Mod(-23,p)));
? (Mod(23,p)^((p+1)/4))^2==Mod(-23,p)
%34 = 1
? p=2^7-1;
? s=lift(sqrt(Mod(-23,p)));
? (Mod(23,p)^((p+1)/4))^2==Mod(-23,p)
%37 = 1
? p=2^5-1;
? s=lift(sqrt(Mod(-23,p)));
? (Mod(23,p)^((p+1)/4))^2==Mod(-23,p)
%40 = 1
?

hermann@7950x:~/llr405src/linux64llr$ for e in 3217 4253 4423 9941 19937 
44497 110503 132049 216091 3021377 6972593
> do
> ./seq $e
> done
Starting Fermat PRP test of 2^3217-1
Using FMA3 FFT length 160, a = 23
2^3217-1 is a Fermat Probable prime (23-PRP)! (969 decimal digits)  Time 
: 8.850 ms.
Candidate saved in file t7628367.inCandidate saved in file ���������
                                                                     for 
further test with Primo.
1
Starting Fermat PRP test of 2^4253-1
Using FMA3 FFT length 192, a = 23
2^4253-1 is a Fermat Probable prime (23-PRP)! (1281 decimal digits)  
Time : 11.036 ms.
Candidate saved in file t7925138.inCandidate saved in file ��������� for 
further test with Primo.
1
Starting Fermat PRP test of 2^4423-1
Using FMA3 FFT length 256, a = 23
2^4423-1 is a Fermat Probable prime (23-PRP)! (1332 decimal digits)  
Time : 12.623 ms.
Candidate saved in file t2957232.inCandidate saved in file ��������G for 
further test with Primo.
1
Starting Fermat PRP test of 2^9941-1
Using AVX-512 FFT length 1K, a = 23
2^9941-1 is a Fermat Probable prime (23-PRP)! (2993 decimal digits)  
Time : 36.387 ms.
Candidate saved in file t9276352.inCandidate saved in file ���������& 
for further test with Primo.
1
Starting Fermat PRP test of 2^19937-1
Using AVX-512 FFT length 1K, a = 23
2^19937-1 is a Fermat Probable prime (23-PRP)! (6002 decimal digits)  
Time : 69.816 ms.
Candidate saved in file t9010301.inCandidate saved in file ���������M 
for further test with Primo.
1
Starting Fermat PRP test of 2^44497-1
Using AVX-512 FFT length 4608, a = 23
2^44497-1 is a Fermat Probable prime (23-PRP)! (13395 decimal digits)  
Time : 378.879 ms.
Candidate saved in file t8327415.inCandidate saved in file ��������ѭ for 
further test with Primo.
1
Starting Fermat PRP test of 2^110503-1
Using AVX-512 FFT length 6K, a = 23
2^110503-1 is a Fermat Probable prime (23-PRP)! (33265 decimal digits)  
Time : 1.143 sec.
1
Starting Fermat PRP test of 2^132049-1
Using AVX-512 FFT length 7K, a = 23
2^132049-1 is a Fermat Probable prime (23-PRP)! (39751 decimal digits)  
Time : 1.576 sec.
1
OK until number 0
OK until number 47293008
Starting Fermat PRP test of 2^216091-1
Using AVX-512 FFT length 10752, a = 23
2^216091-1 is a Fermat Probable prime (23-PRP)! (65050 decimal digits)  
Time : 3.615 sec.
1
Resuming Fermat PRP test of 2^3021377-1 at bit 223552 [7.39%]
Using AVX-512 FFT length 192K, Pass1=3K, Pass2=64, clm=1, 8 threads, a = 
23
2^3021377-1 is a Fermat Probable prime (23-PRP)! (909526 decimal digits) 
 Time : 451.393 sec.
1
Starting Fermat PRP test of 2^6972593-1
Using AVX-512 FFT length 360K, Pass1=960, Pass2=384, clm=2, 16 threads, 
a = 23
2^6972593-1 is a Fermat Probable prime (23-PRP)! (2098960 decimal 
digits)  Time : 1121.495 sec.
1
hermann@7950x:~/llr405src/linux64llr$