[x,y,s] for top 10 known primes p with s²=-1 (mod p) and p=x²+y² (all above 6million decimal digits)

top10 primes "=1 (mod 4)" from largest known primes list

top10.gp

{[                   \\ https://t5k.org/primes/lists/all.txt    as of 9/28/2024
[-1,516693,1048576], \\  7  earlier: Phi(3,-516693^1048576) 11981518 L4561 2023
[-1,465859,1048576], \\  8  earlier: Phi(3,-465859^1048576) 11887192 L4561 2023
[10223,2,31172165],  \\ 11  10223*2^31172165+1               9383761 SB12  2016
[1,1963736,1048576], \\ 17  1963736^1048576+1                6598776 L4245 2022
[1,1951734,1048576], \\ 18  1951734^1048576+1                6595985 L5583 2022
[202705,2,21320516], \\ 19  202705*2^21320516+1              6418121 L5181 2021
[1,1059094,1048576], \\ 21  1059094^1048576+1                6317602 L4720 2018
[1,919444,1048576],  \\ 23  919444^1048576+1                 6253210 L4286 2017
[81,2,20498148],     \\ 24  81*2^20498148+1                  6170560 L4965 2023
[7,2,20267500]       \\ 25  7*2^20267500+1                   6101127 L4965 2022
]}

The enclosing angle brackets are for allowing PARI/GP "readvec()" multiline input only.

Above format is for avoiding difficult parsing in languages other than PARI/GP only.
For PARI/GP unchanged entries from list of largest known primes directly work:

$ gp -q
? Phi(x,y)=polcyclo(x,y);
? P=readvec("top10.2.gp")[1];
? ##
  ***   last result: cpu time 463 ms, real time 501 ms.
? #P
10
? #digits(P[1])
11981518
? 

$ cat top10.2.gp 
{[ \\ https://t5k.org/primes/lists/all.txt as of 2/25/2024
Phi(3,-516693^1048576), \\  7e         11981518 L4561 2023
Phi(3,-465859^1048576), \\  8          11887192 L4561 2023
10223*2^31172165+1,     \\ 11           9383761 SB12  2016
1963736^1048576+1,      \\ 15           6598776 L4245 2022
1951734^1048576+1,      \\ 16           6595985 L5583 2022
202705*2^21320516+1,    \\ 17           6418121 L5181 2021
1059094^1048576+1,      \\ 19           6317602 L4720 2018
919444^1048576+1,       \\ 21           6253210 L4286 2017
81*2^20498148+1,        \\ 22           6170560 L4965 2023
7*2^20267500+1          \\ 23           6101127 L4965 2022
]}
$

files

name	size	what
top10.gp	4KB	ith top10 prime
sqrtm1.gp	47MB	sqrt(-1) (mod p) for ith prime p
sos.gp	45MB	[x,y] for ith prime =x^2+y^2 (x>y>0)

validation

validate.gp, PARI/GP

hermann@7950x:~/top10.sum_of_2_squares.primes$ gp -q < validate.gp 
P=readvec("top10.gp")[1]
  ***   last result: cpu time 0 ms, real time 0 ms.
conversion
  ***   last result: cpu time 355 ms, real time 376 ms.
S=readvec("sqrtm1.gp")[1]
  ***   last result: cpu time 346 ms, real time 394 ms.
validation
1111111111
  ***   last result: cpu time 1,643 ms, real time 1,660 ms.
XY=readvec("sos.gp")[1]
  ***   last result: cpu time 266 ms, real time 286 ms.
validation
1111111111
  ***   last result: cpu time 384 ms, real time 391 ms.
hermann@7950x:~/top10.sum_of_2_squares.primes$ 

validate.py, Python with gmpy2:
TODO

validate.js, Node with BigInt:
TODO

validate.cc (Makefile), C++ with libgmpxx:

hermann@7950x:~/top10.sum_of_2_squares.primes$ make clean
rm -f validate.cc.out
hermann@7950x:~/top10.sum_of_2_squares.primes$ make validate
g++ -O3 validate.cc -lgmpxx -lgmp -o validate.cc.out
./validate.cc.out
1111111111
1111111111
hermann@7950x:~/top10.sum_of_2_squares.primes$ time make validate 2>/dev/null 
./validate.cc.out

real	0m2.543s
user	0m2.405s
sys	0m0.138s
hermann@7950x:~/top10.sum_of_2_squares.primes$ 

runtimes

Proth prime computations were done on AMD 7950X CPU Ubuntu 22.04 with LLR 4.0.5 software of Jean Penné based on (x86 only) gwnum lib, with 16 threads forced onto 8C/16T chiplet0:
http://jpenne.free.fr/index2.html
Yves Gallot pointed out that no computation is needed for the 5 primes of form x^2+1:
https://mersenneforum.org/showthread.php?p=651245#post651245
mersenneforum.org @Neptune showed how to do the Phi(3,_) prime computations superfast:
https://mersenneforum.org/showthread.php?p=651488#post651488

? {
b=516693;
e=1048576;
p=polcyclo(3,-b^e);
s=b^(e*3/2);
[M,V]=halfgcd(s,p);
[x,y]=[V[2],M[2,1]];
}
? ##
  ***   last result: cpu time 347 ms, real time 347 ms.
? x^2+y^2==p
%19 = 1
? s^2%p==p-1
%20 = 1
? #digits(p)
%21 = 11981518
?

rank	description	digits	who	year	runtime	comment
7e	Phi(3,-516693^1048576)	11,981,518	L4561	2023	347ms
8	Phi(3,-465859^1048576)	11,887,192	L4561	2023	358ms
11	10223*2^31172165+1	9,383,761	SB12	2016	6:33:01h
15	1963736^1048576+1	6,598,776	L4245	2022	—	x^2+1
16	1951734^1048576+1	6,595,985	L5583	2022	—	x^2+1
17	202705*2^21320516+1	6,418,121	L5181	2021	4:09:02h
19	1059094^1048576+1	6,317,602	L4720	2018	—	x^2+1
21	919444^1048576+1	6,253,210	L4286	2017	—	x^2+1
22	81*2^20498148+1	6,170,560	L4965	2023	—	x^2+1
23	7*2^20267500+1	6,101,127	L4965	2022	1:59:20h