My colleague Sergey Grishin and I developed an algorithm for finding a symmetric 19-tuplet with a minimum diameter of 252.

This tuple is found using the following pattern

0, 6, 12, 30, 42, 72, 90, 96, 120, 126, 132, 156, 162, 180, 210, 222, 240, 246, 252

In the future, I plan to launch a second Application in the ODLK2025 project, in which we will search for a symmetric 19-tuplet with a minimum diameter of 252 using a different algorithm that is suitable for the BOINC project.

In the meantime, I want to offer all crunchers 52 "endless" programs for searching for this tuple.
The program is, of course, conditionally infinite.
It can be executed, for example, in a year or more, but in a finite time.
And if you run it on a supercomputer, it can be executed quickly.
But I don't have access to a supercomputer, and on a regular computer the program takes a very long time to run, so I call it infinite.

The program is written in PARI/GP language.

Program text

default(parisizemax,10^9);
default(timer,1);
\l 19porc_71_res.txt;

{
\\enter pattern
pt=[0, 6, 12, 30, 42, 72, 90, 96, 120, 126, 132, 156, 162, 180, 210, 222, 240, 246, 252];

w=71;

fs=15716643102160534111758180;

np1=0; print(np1," from number");
np2=0; print(np2," to number");
central=3;

\\ end of data

pl=#pt;
nw=primepi(w);
printf("%d \n",pt);
print("patterns length ",pl);
prs=primes(nw);
period=vecprod(prs);
print(period," period");
vp=vector(np2-np1+1, i, period*(np1-1+i)); lvp=#vp;
printf("search in %d (%.1E) - %d (%.1E)\n",
vp[1],vp[1],vp[lvp]+period,vp[lvp]+period);
cp=vector(central,i,pt[pl\2-central\2+i]);
printf("central %d: %d\n", central,cp);
printf("prove by %d#: ",prs[nw]);print(prs);

vmy=vector(60); pat1=vector(19); pat2=vector(19);

lpr=1;
wd=vector(nw);

for( ip=1,nw,
rip=[];
for( r=1,prs[ip]-1,
for( i=1,pl, if( (r+pt[i])%prs[ip]==0, next(2)));
rip =concat(rip,r) );
if( prs[ip]==71, rip=[1]);
lpr=lpr*#rip;
wd[ip]=rip;
); \\for ip
print(lpr," formulae expected");

k=0;
forvec(v=vector(#wd,i,[1,#wd[i]]), k++;
form=lift(chinese( vector( #wd,j,Mod( wd[j][v[j]], prs[j]) ) ));

\\ начало проверки кортежа
if (form>fs && ispseudoprime(form),
foreach(vp,bpp,
bpt=form+bpp;

if(ispseudoprime(bpt+252),
l=0;
forprime(p=bpt,bpt+252, l++; vmy[l]=p; );
if(l==19,
for(m=2,19, pat1[m]=vmy[m]-vmy[1]; );
pat2=vector(19,i,(pat1[i]==pt[i]));
vlds=vecsum(pat2);
if(vlds>10,
code=fromdigits(pat2[2..18],2);
print(vmy[1],": ",pat1); print("valids=",vlds); print("code=",code); print("number form=",k); print ();
);\\if vlds>
); \\ if l==19
);\\ if ispseudoprime
);\\ foreach
); \\ if form
\\ конец проверки кортежа

);\\ forvec
}

________________________________________________________

From this text we get 52 different programs by changing just one line, highlighted in red.

This is the ruler by which we will change the specified line in the program

[1,2,4,5,6,7,8,9,11,12,13,14,15,17,18,19,20,21,23,24,25,26,27,28,30,31,34,35,36,37,39,40,42,43,45,47,48,49,50,53,54,55,56,58,60,61,63,64,66,67,68,69]; \\ prs= 71; L=52

I made eight programs for myself, and they are running.
In the line you can see eight numbers highlighted in blue, which I used to get the programs.

In the first program (which is published here), the line being changed looks like this
if( prs[ip]==71, rip=[1]);

Then we go along the rule:
for the second program
if( prs[ip]==71, rip=[2]);

for the third program
if( prs[ip]==71, rip=[4]);
and so on.

In the eighth program I have the following line:
if( prs[ip]==71, rip=[9]);

In the ninth program need to insert the following line
if( prs[ip]==71, rip=[11]);

If you have a free computer that works continuously, please take one "endless" program and run it.

To run it, you will need the gp.exe program
You can take this program in one of my archives, for example, here
https://disk.yandex.ru/d/iDTZBVlH5g8yTw

The working program that is published here has the name 19porc_71.txt

The gp.exe program runs the working program in the program shell, the command line to run
gp.exe 19porc_71.txt

You can run several different programs if you have the resources.
But do not run several programs in one folder!

The program will rarely find results, you will see them in the console.
They will also be written to the output file 19porc_71_res.txt
Basically, these will be approximations to the desired tuple.
The exact tuple will be found if valids=19 is written in the solution.

If you have any questions, ask them in this thread or in PM.

Note: if you run and execute all 52 programs, all symmetrical 19-tuplets with a minimum diameter in the given range will be found.
But this is only possible on very powerful equipment - a cluster or a supercomputer.

The search range in this subtask
(15716643102160534111758180, 557940830126698960967415390).

This is the console in one of the programs I'm running

? \r 19porc_71e.txt
  ***   Warning: new maximum stack size = 1000000000 (953.674 Mbytes).
   log = 1 (on)
   [logfile is "19porc_71e_res.txt"]
0 from number
0 to В  number
[0,6,12,30,42,72,90,96,120,126,132,156,162,180,210,222,240,246,252]
patterns length 19
557940830126698960967415390 period
search in 0 (0.E-19) - 557940830126698960967415390 (5.6 E26)
central 3: [120,126,132]
prove by 71#: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71]
293416757467545600 formulae expected
253065723084184442877779701: [0, 6, 12, 30, 42, 58, 78, 82, 108, 126, 136, 160, 178, 198, 210, 222, 240, 246, 252]
valids=11
code=123151
number form=614466877

You see the approximation found

253065723084184442877779701: [0, 6, 12, 30, 42, 58, 78, 82, 108, 126, 136, 160, 178, 198, 210, 222, 240, 246, 252]

The program outputs approximations with valids>10.
Approximations can be with valids: 11, 12, 13, …, 18.
The exact tuple has valids=19.

Recently, after a long search (almost two years), a group of researchers on the dxdy.ru forum found the first symmetrical 19-tuplet with a minimum diameter of 252

9425346484752129657862217: 0, 6, 12, 30, 42, 72, 90, 96, 120, 126, 132, 156, 162, 180, 210, 222, 240, 246, 252

See message
https://dxdy.ru/post1668659.html#p1668659

We need to find many such tuples, from which we can obtain a symmetrical 21-tuplet with a diameter of 360.

This 21-tuplet has the following pattern

0, 54, 60, 66, 84, 96, 126, 144, 150, 174, 180, 186, 210, 216, 234, 264, 276, 294, 300, 306, 360

In the 10 years of the project's existence, not a single symmetrical 21-tuplet has been found!

Announcement

Successive patterns.
Matryoshka dolls.

It will be tomorrow.

See
https://boinc.mak.termit.me/odlk2025/forum_thread.php?id=22

I have a result in another "endless" program

? \r 19porc_71g.txt
  ***   Warning: new maximum stack size = 1000000000 (953.674 Mbytes).
   log = 1 (on)
   [logfile is "19porc_71g_res.txt"]
0 from number
0 to В  number
[0,6,12,30,42,72,90,96,120,126,132,156,162,180,210,222,240,246,252]
patterns length 19
557940830126698960967415390 period
search in 0 (0.E-19) - 557940830126698960967415390 (5.6 E26)
central 3: [120,126,132]
prove by 71#: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71]
293416757467545600 formulae expected
166262574678022506733814131: [0, 6, 12, 22, 42, 52, 72, 78, 108, 126, 132, 142, 162, 180, 210, 222, 232, 246, 252]
valids=12
code=106941
number form=5922702344

You may ask: "What if the program suddenly breaks for external reasons?"

You simply start another program from 52.

I currently have eight programs running all the time.
If they all break (on this computer), I will start eight new programs.

36 "бесконечных" программ из этого пакета давно выполняет termit.

Он периодически присылает мне результаты.
Эти результаты пополняют спектр приближений к 19-ке с минимальным диаметром.
Смотрите тему об этом спектре
https://boinc.mak.termit.me/odlk2025/forum_thread.php?id=48
Там тоже происходит пополнение данного спектра приближений.

Результатов очень много в одной порции, поэтому я обрабатываю их небольшими частями.

Это новые приближения к 19-ке с минимальным диаметром с уникальными кодами, полученные проверкой первой части порции результатов

30013916054020837199065171: [0, 6, 12, 30, 58, 72, 90, 108, 120, 126, 148, 160, 162, 168, 180, 208, 232, 246, 252]
valids=11
code=121633
number form=318355133993

152814314552321450705321011: [0, 6, 12, 52, 70, 72, 78, 90, 96, 120, 132, 156, 162, 168, 178, 222, 240, 246, 252]
valids=11
code=102631
number form=361784444738

103798909526707967826899011: [0, 12, 22, 30, 42, 58, 90, 96, 108, 126, 156, 162, 168, 180, 210, 232, 240, 246, 252]
valids=11
code=27931
number form=346410618673

367759740889668380068940491: [0, 6, 12, 30, 52, 58, 76, 96, 120, 138, 160, 162, 178, 180, 208, 222, 240, 246, 252]
valids=11
code=116247
number form=325985769851

416256098647400379620927041: [0, 6, 22, 30, 48, 82, 90, 96, 120, 126, 136, 156, 160, 168, 210, 222, 240, 246, 252]
valids=13
code=85839
number form=290649083613

17163002887226604401537911: [0, 6, 12, 22, 42, 58, 72, 76, 120, 138, 156, 160, 162, 198, 210, 222, 240, 246, 252]
valids=11
code=107055
number form=327200029832

Спектр содержит 63145 уникальных элементов.

Обработка порции результатов будет продолжена.

Благодарю termit за выполнение этих "бесконечных" программ.
Программы будут выполняться очень долго, пока будет работать компьютер.
Это интересный алгоритм поиска.
Его нельзя было запустить в BOINC-проекте, потому что программы работают слишком долго.

Однако я придумала метод распараллеливания подобных "бесконечных" программ на короткие конечные программы.
Это мой новый алгоритм.
Он пока не реализован в BOINC-проекте.
Мы планируем это сделать в ближайшее время.

---

The TBEG BOINC project maintains the database
https://boinc.tbrada.eu/spt/explore.php

Ещё часть результатов обработала от "бесконечных" программ.

Найдено 8 новых приближений к 19-ке с минимальным диаметром с уникальными кодами

421293461689135876368612451: [0, 6, 12, 30, 48, 78, 90, 100, 120, 126, 138, 160, 162, 180, 210, 222, 240, 246, 252]
valids=14
code=117567
number form=378606760208

30006114856192789760540461: [0, 6, 12, 42, 52, 72, 78, 96, 120, 126, 132, 138, 162, 180, 196, 222, 232, 240, 252]
valids=12
code=104372
number form=339673547717

275413889476874622907567501: [0, 22, 30, 42, 70, 72, 76, 108, 120, 126, 132, 156, 162, 180, 196, 222, 232, 246, 252]
valids=11
code=5109
number form=316222197384

278943042059380835100078481: [0, 6, 12, 30, 58, 70, 82, 96, 108, 126, 132, 156, 168, 178, 198, 222, 240, 246, 252]
valids=12
code=116167
number form=385317325180

416519584381673970494821891: [0, 6, 12, 30, 52, 70, 78, 96, 120, 126, 136, 156, 160, 178, 210, 222, 240, 246, 252]
valids=13
code=116559
number form=386883618386

535315312864527156786188731: [0, 6, 30, 42, 70, 72, 78, 96, 120, 126, 132, 142, 156, 198, 210, 222, 240, 246, 252]
valids=12
code=71567
number form=336675460059

262820198587904204188135561: [0, 6, 12, 30, 48, 82, 90, 96, 120, 126, 132, 138, 162, 168, 198, 222, 240, 246, 252]
valids=14
code=118695
number form=317790598718

254943010932964916966528491: [0, 6, 22, 30, 42, 72, 78, 96, 108, 126, 132, 178, 180, 208, 210, 222, 240, 246, 252]
valids=13
code=95631
number form=279267116476

Спектр содержит 63155 уникальных элементов.

---

The TBEG BOINC project maintains the database
https://boinc.tbrada.eu/spt/explore.php

Corporal прислал очередную порцию результатов от 36 "бесконечных" программ.

Начала обрабатывать.
Вот первые восемь приближений к 19-ке с минимальным диаметром с уникальными кодами из этой порции

543696882204036286545433981: [0, 6, 12, 30, 58, 78, 82, 96, 126, 138, 160, 168, 178, 180, 210, 222, 240, 246, 252]
valids=11
code=115743
number form=476180517489

59072592142185078585773011: [0, 6, 12, 30, 48, 72, 76, 78, 96, 126, 132, 136, 162, 180, 190, 198, 222, 246, 252]
valids=11
code=119217
number form=468941391436

299550142902319322140291531: [0, 6, 12, 30, 48, 72, 78, 90, 96, 126, 142, 160, 162, 180, 190, 196, 240, 246, 252]
valids=11
code=119091
number form=476710156060

98429173227795768506863081: [0, 6, 12, 30, 42, 48, 76, 90, 96, 126, 132, 156, 198, 208, 210, 222, 232, 240, 252]
valids=11
code=123340
number form=460034685409

539890979284841290360488601: [0, 6, 12, 30, 72, 78, 82, 96, 120, 138, 148, 156, 162, 168, 208, 222, 240, 246, 252]
valids=12
code=116327
number form=451926906534

502114085701462527292834201: [0, 6, 12, 30, 76, 82, 90, 96, 120, 132, 162, 196, 198, 208, 210, 232, 240, 246, 252]
valids=11
code=118283
number form=445242240546

17945556825059518919999101: [0, 6, 30, 48, 70, 72, 90, 96, 120, 126, 136, 138, 168, 196, 198, 222, 240, 246, 252]
valids=11
code=73479
number form=480346232977

119078380967585692044106081: [0, 6, 12, 42, 58, 78, 90, 100, 120, 132, 148, 156, 162, 180, 190, 196, 240, 246, 252]
valids=11
code=100979
number form=476852920305

Спектр содержит 63214 уникальных элементов.

Завтра продолжу обработку.

---

The TBEG BOINC project maintains the database
https://boinc.tbrada.eu/spt/explore.php

Ещё семь приближений к 19-ке с минимальным диаметром от "бесконечных" программ

326007155905160949000165751: [0, 6, 22, 52, 70, 72, 76, 82, 108, 126, 132, 156, 168, 190, 210, 222, 240, 246, 252]
valids=11
code=70095
number form=392966091560

428601800435143718013605341: [0, 6, 12, 30, 42, 72, 76, 78, 90, 100, 126, 156, 180, 196, 210, 232, 240, 246, 252]
valids=11
code=127051
number form=396766159541

538158896183605906696501381: [0, 6, 22, 30, 48, 72, 90, 96, 120, 126, 132, 148, 156, 180, 198, 210, 222, 246, 252]
valids=12
code=90001
number form=441459733514

432718180098234045515444311: [0, 6, 22, 30, 42, 70, 76, 108, 120, 126, 132, 136, 180, 198, 210, 222, 240, 246, 252]
valids=12
code=91023
number form=417534613769

33660305541563151973286131: [0, 52, 58, 70, 72, 82, 90, 96, 108, 126, 132, 138, 162, 180, 198, 222, 240, 246, 252]
valids=11
code=3511
number form=426674653930

230629643698635229578304141: [0, 6, 12, 30, 48, 52, 90, 108, 132, 138, 148, 156, 160, 180, 210, 222, 232, 246, 252]
valids=11
code=116829
number form=418192052404

433163847683917290515999791: [0, 6, 12, 30, 52, 72, 78, 108, 126, 138, 156, 160, 178, 180, 210, 222, 240, 246, 252]
valids=11
code=118815
number form=392423506165

Спектр содержит 63224 уникальных элемента.

Завтра продолжу обработку.

---

The TBEG BOINC project maintains the database
https://boinc.tbrada.eu/spt/explore.php

Последние 15 приближений к 19-ке с минимальным диаметром с уникальными кодами от "бесконечных" программ

220279065019059828238050961: [0, 6, 12, 30, 42, 70, 76, 78, 120, 132, 138, 156, 180, 198, 210, 222, 240, 246, 252]
valids=12
code=123471
number form=469894273593

169682775248774906835561211: [0, 6, 12, 30, 42, 48, 52, 108, 120, 126, 132, 136, 168, 180, 210, 222, 240, 246, 252]
valids=14
code=123807
number form=466623923846

225881711600380722799657111: [0, 6, 42, 52, 72, 76, 90, 108, 120, 126, 132, 156, 178, 180, 210, 222, 232, 246, 252]
valids=12
code=68573
number form=415142026989

57708666609705132977059471: [0, 6, 12, 30, 42, 48, 76, 96, 100, 126, 136, 156, 160, 180, 196, 232, 240, 246, 252]
valids=12
code=124243
number form=453313480070

55552010227799913425892661: [0, 6, 12, 30, 42, 76, 90, 100, 120, 126, 136, 138, 180, 190, 222, 232, 240, 246, 252]
valids=11
code=125699
number form=496472358187

61925039188384485432653701: [0, 6, 22, 30, 70, 72, 90, 96, 100, 126, 132, 156, 162, 168, 178, 190, 198, 222, 252]
valids=11
code=89568
number form=487681295516

527180571831356208337085881: [0, 6, 22, 30, 48, 58, 70, 108, 120, 126, 136, 156, 162, 208, 210, 222, 240, 246, 252]
valids=12
code=82799
number form=418254785296

474783978414356885833092901: [0, 6, 12, 42, 70, 72, 82, 100, 120, 126, 132, 156, 178, 180, 190, 222, 232, 246, 252]
valids=12
code=103381
number form=467596833740

456596714042470370679074131: [0, 6, 12, 30, 42, 76, 78, 82, 120, 126, 132, 148, 178, 180, 210, 222, 232, 246, 252]
valids=13
code=123805
number form=420937777943

517556828496867103578758491: [0, 6, 22, 30, 42, 72, 90, 120, 138, 142, 148, 156, 160, 180, 196, 222, 240, 246, 252]
valids=12
code=96343
number form=389049252361

118053400296155027856706921: [0, 6, 12, 30, 48, 76, 78, 96, 120, 132, 138, 156, 178, 190, 208, 222, 240, 246, 252]
valids=11
code=116295
number form=440329433792

511333932854456496358288081: [0, 12, 22, 30, 42, 72, 90, 96, 120, 126, 156, 162, 178, 180, 198, 222, 240, 246, 252]
valids=13
code=32535
number form=415349319448

290824654727189923262386831: [0, 6, 22, 48, 70, 72, 90, 96, 120, 148, 160, 162, 178, 180, 196, 222, 240, 246, 252]
valids=11
code=73239
number form=413178975194

316220462038002809467905091: [0, 6, 12, 30, 42, 76, 90, 96, 100, 142, 148, 156, 178, 196, 210, 222, 232, 246, 252]
valids=12
code=126029
number form=407372687206

48183429203479616233471651: [0, 6, 12, 58, 70, 78, 90, 96, 108, 120, 132, 160, 162, 168, 210, 222, 232, 246, 252]
valids=11
code=101549
number form=487296339430

Спектр содержит 63239 уникальных элементов.

Самое интересное приближение с valids=14

169682775248774906835561211: [0, 6, 12, 30, 42, 48, 52, 108, 120, 126, 132, 136, 168, 180, 210, 222, 240, 246, 252]
169682775248774906835561211: [0, 0, 0, 0, 0, -24, -38, 12, 0, 0, 0, -20, 6, 0, 0, 0, 0, 0, 0]
169682775248774906835561211: [1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1]
valids=14
code=123807

В этом приближении содержатся приближение к ключевой 17-ке, к центральным 15-ке, 13-ке и 11-ке.
Коды приближений не уникальные.

---

The TBEG BOINC project maintains the database
https://boinc.tbrada.eu/spt/explore.php