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

On Sun, Nov 17, 2024, 5:55 AM <hermann@stamm-wilbrandt.de> wrote:

On 2024-11-17 03:08, Kurt Foster wrote:
That is, -23 is a quadratic residue (mod M).
> Using the usual formula, we have that
>
> (-23)^((M+1)/4) is a square root of -23 (mod M). [Since (M+1)/4 is
> even (in fact, a power of 2), we can take 23^((M+1)/4) instead.]
>
> However, this does NOT insure that M = x^2 + 23*y^2. One possible way
> to test the question is to compute a square root of -23 (mod M) and
> use halfgcd (Cornacchia algorithm) and see whether it produces a
> solution.
>
> 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.
>
Does your argument apply to the quadratic residue -3 for Mersenne primes
as well?

What is different for prime p=1 (mod 4) where we know that -1 is
quadratic residue AND that always integers x,y exists with p=x^2+y^2?

Q(i) has class number 1. There are two things to worry about. The first is the ring of integers will be bigger in Q(sqrt(d)) than just Z[sqrt(d)]. Thankfully we know what it is but we might get 4p out instead of p.

Secondly the class number might be larger as happens for 41 and -5. One of the ideals has a generator with norm 41, the other does not.

Thankfully we know the complete theory of representability by binary quadratic forms.

I "easily" can run the test you proposed for M_52, but as the last run
that took 8 days with 1580% CPU, the new run would take the same time.

So I looked for which Mersenne primes -23 is quadratic residue:

pi@raspberrypi5:~ $ gp -q mpe.gp
? c=1;foreach(mpe,e,if(kronecker(-23,2^e-1)==1,print1(c,","));c+=1)
1,3,4,5,7,10,11,14,18,19,20,22,24,27,29,30,31,37,38,39,42,43,44,46,47,49,51,52,
?

Then I chose M_31 because M_30 did not do Fermat PRP test.
I used the patched software that writes out the "M_31-2"th value
making use of LLR software Fbase option:

hermann@7950x:~/llr405src/linux64llr$ taskset -c 0-7,16-23 ./sllr64
-oFbase=23 -t16 -d -q"2^216091-1" 2>err
Starting Fermat PRP test of 1*2^216091-1
Using AVX-512 FFT length 10752, a = 23
1*2^216091-1 is a Fermat Probable prime (23-PRP)! (65050 decimal digits)
Time : 3.630 sec.
...

I copied the first value ...

hermann@7950x:~/llr405src/linux64llr$ grep 216089: err | cut -b-60 |
head -1
\\ 216089: 0x05690487136e2724b85c71e5df0a60d2e4341edb4b6994d
hermann@7950x:~/llr405src/linux64llr$

... into this GP script:

hermann@7950x:~/llr405src/linux64llr$ cut -b-60 M_31.23.gp
p=2^216091-1;
s=0x05690487136e2724b85c71e5df0a60d2e4341edb4b6994d1604afe2f
Mod(s,p)^2==Mod(-23,p)
hermann@7950x:~/llr405src/linux64llr$

And we have found sqrt(Mod(-23,M_31)):

hermann@7950x:~/llr405src/linux64llr$ gp -q < M_31.23.gp
1
hermann@7950x:~/llr405src/linux64llr$

Script size is not too big, so you can find script below:

hermann@7950x:~/llr405src/linux64llr$ wc --bytes M_31.23.gp
54067 M_31.23.gp
hermann@7950x:~/llr405src/linux64llr$

Regrds,

Hermann.

hermann@7950x:~/llr405src/linux64llr$ cat M_31.23.gp
p=2^216091-1;
s=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db4016ec613d230b139402a3a8b87f9903a60d80621ba544805d3bcf810f71df07fb20a7930390ed023df73f1afff9bb990f10931130438adcf7323e31c453fbb3006d35569a4556ba7ec1ac5e6d0f9f974ffd178d9a3de5823c8d80e3bc6eb9f87fb44b9a8ccf62847a236cfed492589b15f25280060d2762aa18db22e5ec0e4833427674866a6e8b96a17b34fb2243152db18d483c64a81d4dd5bad43b232ef62c176df74249e0b4a023d13127214d8598890e4d1e04cc1c99a86526739b3be96baa03582d1d774bc22c9d4871b52c746e4df8b221214fdcf34e89531ad250017d730b81375cefd6f95aaaee141c5af3912995489f6309d65acbea2c5007f99cda47bfbe9691e7d2e8e13121f79f56fa4e177d5a89bb41e36fdb2d0ca25d9a55bf2c6c0534f981faaf512be65e06d4a10d86a8a254acff1e8b5aa1b79355cb92886f25176ede1555ddf599a823e469d2bd9ebfdb62d9fd6b93d49bdfcdae4ef8ed3d09ebc4dfb4224c34ead89a5a51c19025e23a586f148582493a70e78dd09dbcf6d06b30ffef643c28496e02778848ff398bbcbd062e9c34a04865d395dcc8e701da4d3dd9c66e116f164dee997d5485f8f26035462259fd22e81016adb6651116ae7c4d4d2f1696e236a794344038d520d520a66241250dae93981ab971930be018f739ece6d3eb651a9966707c06a9c3
e0504a228a8aa664e52684a3a5cee76b353d08d0a318014f95a717f26e1a08c14a2669dbadceae6bf9a9011c4e689969fbe872a21e645721743f7f5a7164462f59f42c13;
Mod(s,p)^2==Mod(-23,p)
hermann@7950x:~/llr405src/linux64llr$