new prime ktuplet page
Hello members !
Tony Forbes told me, from sep. 21 is the old "prime ktuplet" page frozen by google. He ask me for taking over his work. Yes, today I bought a new domain. I will [U]try[/U] to carry on his layout. New link: [URL="http://www.pzktupel.de\ktuplets.htm"]www.pzktupel.de\ktuplets.htm[/URL] So all future results or records to me. best wishes Norman 
Nice one, good luck!
A small observation, the term "first appearance", used a lot there, is somehow confuse (and misleading), for sure most of those are not "first appearances" (i.e. there are a lot of smaller primes of the same size, for almost all those in the tables). It seems you use the term as the earliest date of discovery, or something. All those cases should bear an asterisk or footnote saying so. 
first appearances
You mean "first known appearances" is better ?
"first appearances" is take over 
Could you add details on software that can be used (and how to use it) for finding tuplets? I think that many people would be interested in finding new records. A few might actually be interested in improving the speed of the software used to find them.

[QUOTE=Cybertronic;586116]You mean "first known appearances" is better ?
"first appearances" is take over[/QUOTE] No, I mean it should say like "sorted by the time of discovery" or something. Some native English speaker can weight in better than me. For example, for 100 digits, there are lots and lots of primes smaller than 2^5211. That can not be "first apparition". Even for 157 digits itself, this is not the "first appearance", that should be 10^156+451, which can be found in 10 milliseconds with pari (type: nextprime(10^156)). Maybe it was the earliest discovered, but that's also arguable. The "first apparition" of twin primes of 100+x digits, for x=0 to 9 can be found very fast with nowadays computers, the next script runs just 7 seconds for 100 to 110, single core (and similar, for larger tuples, for so low digits count), so it would not take too long to solve the "unreliable" part of the tables. What is "unreliable" about?? [CODE] firstTwins(fromDigit=1, toDigit=200)= { for(n=0,toDigitfromDigit, q=1; forprime(p=10^(fromDigit+n1),10^(fromDigit+n), if(pq == 2, if(q == 3, print("Found 1 digit: 10^0+3 + {0,2}"), print("Found "fromDigit+n" digits: 10^"fromDigit+n1"+"q10^(fromDigit+n1)" + {0,2}") ); break ); q = p ) ); } gp > firstTwins(100,110) Found 100 digits: 10^99+6001 + {0,2} Found 101 digits: 10^100+35737 + {0,2} Found 102 digits: 10^101+139201 + {0,2} Found 103 digits: 10^102+106759 + {0,2} Found 104 digits: 10^103+29659 + {0,2} Found 105 digits: 10^104+3457 + {0,2} Found 106 digits: 10^105+113617 + {0,2} Found 107 digits: 10^106+94789 + {0,2} Found 108 digits: 10^107+52819 + {0,2} Found 109 digits: 10^108+66517 + {0,2} Found 110 digits: 10^109+35371 + {0,2} gp > ## *** last result computed in 6,876 ms. gp > nextprime(10^156)10^156 % = 451 gp > firstTwins(157,157) Found 157 digits: 10^156+10489 + {0,2} gp > [/CODE] 
Gute Arbeit!:bow:
[QUOTE=Cybertronic;586116]You mean "first known appearances" is better ? "first appearances" is take over[/QUOTE] I'd vote for "earliest discovery of 100 (1000 etc.) digits". Then you may add "first appearances of ... digits", as you have also calculated them, as well, maybe with a link to your detailed list "smallestndigitprimektuplets". 
[QUOTE=mart_r;586183]Gute Arbeit!:bow:
I'd vote for "earliest discovery of 100 (1000 etc.) digits". Then you may add "first appearances of ... digits", as you have also calculated them, as well, maybe with a link to your detailed list "smallestndigitprimektuplets".[/QUOTE] I will not change the wording of T. Forbes, but I have extended under "27." a link to the smallest ktuplet session. [url]http://www.pzktupel.de/ktuplets[/url] BTW, today I found the second kind of "smallest googol prime 10tuplet". So the pair is now known. 
Early discovery
Okay, I changed it now into "Early discovery"
and took other colors. [url]http://www.pzktupel.de/ktuplets[/url] I hope it is correct now. Norman 
Nice. I love the "smallest tuples" list, the only observation is that you should put the "last updated" text at the beginning, and not at the end. That's for us, so you won't force the reader to go to the end (when I open the page I should see immediately if there was any update, and don't waste my time), but also [B][U]for you[/U][/B] :razz: so you won't forget changing the date when you make updates, hihi (the 2012 can't be right as long as you have tuples discovered in 2013, 2014, etc).
:tu: 
>> "last updated" text at the beginning, and not at the end.
>> the 2012 can't be right... @LaurV Where exactly ? 
See? Told you! It is difficult to find even for yourself :razz: (I just searched for "smallest" to find it again)
[URL="http://www.pzktupel.de/ktmin.txt"]Here[/URL], linked from [URL="http://www.pzktupel.de/ktuplets"]here[/URL], section 23, "Odds and Ends". (I didn't go through all the site yet, but that will come, trust me, hehe). [STRIKE]And as we spoke of it, maybe you can add section links, you know, "#something" like wiki has, so we don't need to specify the section and chapter when we talk about it.[/STRIKE] scratch that, you have that already, but I am silly. Very good work! 
I understand.
I added my smallest titanic quintuplet to pattern d=0,4,6,10,12 and set on today's date on top of the list. [URL]http://www.pzktupel.de/ktmin.txt[/URL] okay ? 
I extended now the page with point 28. (Archive of smallest prime ktuplets)
I try to calculate with my code the first 1000000 initial members for some ktuplets. twins are done. 
k=2,3,4,(5) done
[URL]http://www.pzktupel.de/smarchive.html[/URL] OEIS have 10000 or less 
Update
First 1,000,000 initial prime ktuplets are availabe for k=2,3,4,5,6,7 and each pattern.
[url]http://www.pzktupel.de/smarchive.html[/url] greetings 
k=8
First 1,000,000 initial prime ktuplets are availabe for k=8 and each pattern.
[URL]http://www.pzktupel.de/smarchive.html[/URL] 
k=9
Update:
The first 1000000 members of each prime 9tuplet pattern is now available. 
Hi All,
Thank you for your efforts, Norman. I appreciate your work for ktuples web page. I made a similar web page (that I can no longer edit, and it was free to me) see [URL="https://sites.google.com/site/primeconstellations/"]https://sites.google.com/site/primeconstellations/[/URL] Regards, Matt 
...[COLOR=teal]
[/COLOR] 
[QUOTE=MattcAnderson;587375]Hi All,
Thank you for your efforts, Norman. I appreciate your work for ktuples web page. I made a similar web page (that I can no longer edit, and it was free to me) see [URL]https://sites.google.com/site/primeconstellations/[/URL] Regards, Matt[/QUOTE] Thank you MattcAnderson [COLOR=Black]![/COLOR] I quarried also on your webpage in the past. >I can no longer edit, and it was free to me [COLOR=teal][COLOR=Black]Yes, like Tony. [/COLOR][/COLOR][COLOR=teal][COLOR=Black]How long is the page visible ?[/COLOR][/COLOR] [COLOR=teal][COLOR=Black] [/COLOR][/COLOR] [COLOR=teal][COLOR=Black]For me it costs few Euro/Dollar per month, [/COLOR][/COLOR][COLOR=teal][COLOR=Black]but to me it is worth it. [/COLOR][/COLOR] [COLOR=teal][COLOR=Black](before all informations are lost) [/COLOR][/COLOR] [COLOR=teal][COLOR=Black]Norman[/COLOR][/COLOR] 
Hi Norman
I have no idea how long Google will make my Prime Constellations web page visible. I backed it up for archive purposes, but don't have an immediate plan for sharing it online. Feel free to copy or share the information in that web page as you want. Regards, Matt 
The most links are intact , alle pdf's I downloaded.
I think , it is not much to tie in the page. But the [SIZE=2]"Data for finding Constellations" I will reset by a link to my pattern table. [/SIZE] 
Hi all,
Thanks to Norman Luhn, for copying my Prime Constellations web page and hosting it on the internet. My stuff is one of the links at this address [url]http://www.pzktupel.de/Links.html[/url] Regards, Matt Anderson PS Duplicate information can be found at mattanderson.fun 
k=10
Update: The first 1,000,000 initial members for each prime 10tuplet are available.
[url]http://www.pzktupel.de/smarchive.html[/url] 
k=11
First 100,000 initial members of prime 11tuplets and each pattern available. (up to 3,84e19)
[URL]http://www.pzktupel.de/smarchive.html[/URL] 
k=12
First 20000 initial prime 12tuplets available.
[url]http://www.pzktupel.de/smarchive.html[/url] greetings 
new section Pi_k(X)
Update: Under point 28 are now values of Pi_k(X) ,k=1..12
[url]http://www.pzktupel.de/ktuplets[/url] 
k=13
The first 5000+ initial members to each prime 13tuplet pattern now available.
See: [url]http://www.pzktupel.de/smarchive.html[/url] 
PI_3(10^15)
Value of 10^15 for prime triplets is known.
[URL]http://www.pzktupel.de/counting/PI_03.html[/URL] Time to compute: 1d 
That's some especially nice data. Have you considered updating A055737, A125517, A063501 etc.? (The latter was one of my first submissions to OEIS. Back in the day, I was excited to have a program to count the sextuplets up to 10^10  or was it 10^9?  within less than two hours.)

A063501
Hello Martin, I updated A063501.

Addition:
I found few values of PI_3(X) and PI_4(X) here. End of page... [url]https://faculty.lynchburg.edu/~nicely/counts.html#Triplets[/url] So my results are correct :smile: Also I added now any names of searchers.... 
Hi all,
Norman is doing a great project!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Matt 
[QUOTE=MattcAnderson;590487]Hi all,
Norman is doing a great project!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Matt[/QUOTE] :bow: At the moment , I calculate PI_14(10^22) It tooks only 4 days per pattern plus any breaks. Norman P.S. Using PCs [2 Ryzen 7 1700 3 GHz] 
PI_14(10^22)
PI_14(10^22) is done.
There are 1810 14tuplets up to 10^22 [URL]http://www.pzktupel.de/Tables.html[/URL] next: PI_15(10^22) 
x congruent 29#
I updated the pattern list for prime 25..30 tuplets.
Prime 30tuplets are the first case of x congruent 29# [URL]http://www.pzktupel.de/ktpatt.html[/URL] This is only a theoretical fact. pattern : 0 4 6 10 16 18 28 30 34 36 48 58 60 64 66 70 76 78 84 88 94 100 106 108 114 118 120 126 130 136; first number : 36990193 (modulo 223092870) pattern : 0 6 10 16 18 22 28 30 36 42 48 52 58 60 66 70 72 76 78 88 100 102 106 108 118 120 126 130 132 136; first number : 186102541 (modulo 223092870) 
Fabelhaft! :bow:
[QUOTE=Cybertronic;591212] [URL]http://www.pzktupel.de/ktpatt.html[/URL] [/QUOTE] Call me naive, but I believe the first numbers of 6tuplets can be restricted to 97 mod 210... have only looked at the first few examples though. 
[QUOTE=Cybertronic;591197]PI_14(10^22) is done.
There are 1810 14tuplets up to 10^22 [URL]http://www.pzktupel.de/Tables.html[/URL] next: PI_15(10^22)[/QUOTE] I saw your page and I have a question: How many classes of prime ntuplets? [CODE] n classes 1 {0} (1 class) 2 {0,2} (1 class) 3 {0,2,6}, {0,4,6} (2 classes) 4 {0,2,6,8} (1 class) 5 {0,2,6,8,12}, {0,4,6,10,12} (2 classes) 6 {0,4,6,10,12,16} (1 class) 7 {0,2,6,8,12,18,20}, {0,2,8,12,14,18,20} (2 classes) 8 {0,2,6,8,12,18,20,26}, {0,6,8,14,18,20,24,26}, {0,2,6,12,14,20,24,26} (3 classes) 9 {0,2,6,8,12,18,20,26,30}, {0,4,6,10,16,18,24,28,30}, {0,2,6,12,14,20,24,26,30}, {0,4,10,12,18,22,24,28,30} (4 classes) 10 {0,2,6,8,12,18,20,26,30,32}, {0,2,6,12,14,20,24,26,30,32} (2 classes) [/CODE] and I searched "[URL="https://oeis.org/search?q=1%2C+1%2C+2%2C+1%2C+2%2C+1%2C+2%2C+3%2C+4%2C+2&language=english&go=Search"]1, 1, 2, 1, 2, 1, 2, 3, 4, 2[/URL]" in OEIS, and no result about prime ktuple found. 
Thanks Martin, this was not optimal. Overlooked !:smile:
[QUOTE=mart_r;591254]Fabelhaft! :bow: Call me naive, but I believe the first numbers of 6tuplets can be restricted to 97 mod 210... have only looked at the first few examples though.[/QUOTE] 
[QUOTE=sweety439;591255]I saw your page and I have a question: How many classes of prime ntuplets?
[CODE] n classes 1 {0} (1 class) 2 {0,2} (1 class) 3 {0,2,6}, {0,4,6} (2 classes) 4 {0,2,6,8} (1 class) 5 {0,2,6,8,12}, {0,4,6,10,12} (2 classes) 6 {0,4,6,10,12,16} (1 class) 7 {0,2,6,8,12,18,20}, {0,2,8,12,14,18,20} (2 classes) 8 {0,2,6,8,12,18,20,26}, {0,6,8,14,18,20,24,26}, {0,2,6,12,14,20,24,26} (3 classes) 9 {0,2,6,8,12,18,20,26,30}, {0,4,6,10,16,18,24,28,30}, {0,2,6,12,14,20,24,26,30}, {0,4,10,12,18,22,24,28,30} (4 classes) 10 {0,2,6,8,12,18,20,26,30,32}, {0,2,6,12,14,20,24,26,30,32} (2 classes) [/CODE] and I searched "[URL="https://oeis.org/search?q=1%2C+1%2C+2%2C+1%2C+2%2C+1%2C+2%2C+3%2C+4%2C+2&language=english&go=Search"]1, 1, 2, 1, 2, 1, 2, 3, 4, 2[/URL]" in OEIS, and no result about prime ktuple found.[/QUOTE] Try one term less in the search and you'll find A[OEIS]083409[/OEIS]. 
Helle sweety439 !
> I saw your page and I have a question: How many classes of prime ntuplets? The exact class for a prime ntuplet I had take over from Tony Forbes. What you mean ? Here is another list: [url]http://www.opertech.com/primes/ktuples.html[/url] Helpfully? 
correction #36. It is 23#, not 29#
:rolleyes:
[Prime 30tuplets are the first case of x congruent 23#] And so on... 
up to prime 50tuplet
The congruentcalculation for all patterns up to prime 50tuplet is done.
See: [url]http://www.pzktupel.de/ktpatt.html[/url] The largest modulonumber is (modulo 10555815270) I hope it is correct. 
new color scheme
[url]http://www.pzktupel.de/ktuplets[/url]

PI_15(10^22)
It tooks me 3 weeks to calculate all prime 15tuplets up to 10^22.
There are 676 exists. Up to 10^21, no missing number was found. [url]https://oeis.org/A257169[/url] 
PI_k(X)
A "Recent additions" for PI_k(X)tables was added
[URL]http://www.pzktupel.de/Tables.html[/URL] 
PI_7(10^17)
Is done after 1day.
There are 93,940,829 7tuplets (both pattern) up to 10^17. [url]http://www.pzktupel.de/counting/PI_07.html[/url] Next PI_6(10^17) Estimate time: 60h 
PI_6(10^17)
It's done ,calculation tooks me 16h.
There are 570,735,178 up to 10^17 [URL]http://www.pzktupel.de/counting/PI_06.html[/URL] Sieving up to ~751000 then RabinMillerTest with base=2. Final result was reduce by 3, I found with the pseudoprime base_2  list (up to 2^64) 3 false prime sextuplets. 12202902616309133 6th number 12658530383462153 6th number 12893382792577873 3rd number 
new section added
[URL]http://www.pzktupel.de/Tables10X.html[/URL]
This table show, how many prime ktuplets are up to 10^n Note, maby PI_6(10^17) ist not correct....must see, why 
:rant:
Bug found, Pi_6(10^17) is done tomorrow. Upper limit was not 10^17,it was ~ 9.9879e16 
PI_6(10^17) correction
The calculation is done: There are 571314626 prime sextuplets up to 10^17.
The sum was reduce by 3 (pseudo prime 6tuplets). Numbers are 12202902616309117+16 12658530383462137+16 12893382792577867+6 PI_6(10^17)=571314626 LI_6(10^17)=571290398 Looks better now :smile: Next: PI_5(10^16) .... estimated time 21h per pattern. 
PI_5(10^16)
A first appoximation after 1h gave me for PI_5(10^16) and pattern d=0,2,6,8,12:
53175713/32*899+PI_5(10^15) ~ 1736450170 LI_5(10^15)=1736513588 Tomorrow, I will see what the real value is. 
PI_5(1e16) pattern d=0,2,6,8,12
Value is 1,736,614,143 and LI=1,736,513,588
Sieving up 8,960,453 There are 10 false prime quintuplet with factors > 8,960,453 1414744276484177, 2043183817242019 ,2591106911415857 ,2762007330795523 ,3650579292309623 ,5459403435729829 ,6612111624056921 6829219377998653 ,6949430519779333 ,7595018731158409 , 2nd pattern done in 12h There will be 15 false prime quintuplets 1186031063218877, 1199012316059017, 3031646959069873, 5138327769220273, 5156031002809789, 5596084581347179, 5857980051959809 6860390003895949, 6944760326240833, 7060112777135767, 7180054501495873, 8492448456384889, 8861685807262037, 8962939623342589 9511159909417093 
PI_5(1e16) pattern d=0,4,6,10,12
PI_5(1e16)=1,736,521,682, for 2nd pattern
So there are 3,473,135,825 prime quintuplets up to 10^16. Next PI_8(10^18) for d=0,2,6,8,12,18,20,26 Estimated time: 18h 
Update PI_8(10^18)
[url]http://www.pzktupel.de/counting/PI_08.html[/url]
3rd pattern is done tomorrow 
PI_8(10^18)
There are 119,309,363 prime octuplets up to 1e18.
Next: PI_9(1e18). Estimated time ~2d for all pattern. 
PI_9(1e18)
There are 13,505,546 prime 9tuplets up to 10^18
[url]http://www.pzktupel.de/counting/PI_09.html[/url] Alle tuplets are uploaded. 
PI_10(10^20)
Next: PI_10(10^20) and also PI_11(10^20)

Update
Pattern list and HardyLittlewood constants are now in one list.
[url]http://www.pzktupel.de/ktpatt_hl.html[/url] 
History
It is a lot to do to refresh the prime ktuplet history.
1997 and 1998 are done! [url]https://www.pzktupel.de/ktuplets.htm[/url] 
prime ktuplet history
The history is now complete refreshed year by year.
[url]https://www.pzktupel.de/ktuplets.htm[/url] best 
New Titanic Prime Sextuplet
Congratulation to Vidar Nakling for a new record prime sextuplet !!!
[url]https://www.pzktupel.de/ktuplets.htm#largest6[/url] It is also a new SP6 record ! [url]http://www.pzktupel.de/JensKruseAndersen/simultprime.htm[/url] 
[QUOTE=Cybertronic;593572]It is also a new SP6 record !
[url]http://www.pzktupel.de/JensKruseAndersen/simultprime.htm[/url][/QUOTE] I scanned through simultprime page and I see (*near the bottom) that this is simply a union of five disparate classes. Many recent CPAPs are missing from this page; for uniformity it is probably better to inject them in appropriate positions. 
?
Give me an example ,please!

[QUOTE=Cybertronic;593578]Give me an example ,please![/QUOTE]
See [URL="http://www.pzktupel.de/JensKruseAndersen/simultprime.htm#history5"]section "5"[/URL]. What CPAPs do you see? Then see [URL="https://www.pzktupel.de/JensKruseAndersen/simultprime.htm#history6"]section "6"[/URL]. What CPAPs do you see? From what year are they? ...Repeat for all sections. 
!
Maybe I understand you , but Jens updated only new records in timeline.
Well, the 1st CPAP6 is larger than the 10th CPAP 5...you mean this ? (should I do it)? 
Well, if it is maintained in this way, then only Table 1 from it is meaningful, really, because the rest is prone to a lot of errors of omission.
For example if you forget to put the 6Tuplet there (currently it is [B]not[/B] there), and then someone beats in tomorrow (but this one was never entered by maintainer), then it will be as if it never happened, because later there is no reason to add it. It would have been more robust to maintain top10s for each "k"length. (But even then it is a very awkward page,  you can compare it to keeping long jump, high jump, triple jump, and discus throwing records in one table. For a concrete example searching for CC's is nearly trivial because both: 1) off the shelf sieve exists, 2) testing is faster than for the other categories. So if someone actually put equal effort in all five categories, then [B]all [/B]records would be obviously only CC's). I'll get done with CPAPs and I will help you "update by way of trivializing" this page. All records will be of the same kind (obviously not CPAPs. :rolleyes: I liked CPAPs because they are hard. Easy things are boring.) 
...
You mean, the SPTable over all is okay, but I should extened a Top10(20) list for each SPkind ,like CC x 1st ans 2nd kind, BiTwin, CPAP,...
Problem is, that is not my page....and nobody reach Jens K Andersen. Okay,the prime ktuplet page have now all kinds and was refreshed yesterday. [URL]https://www.pzktupel.de/ktuplets[/URL] 
simultprime
I have refreshed Jens K Andersens " Simultaneous Primes Prime Page"
[URL]http://www.pzktupel.de/JensKruseAndersen/simultprime.htm[/URL] I hope it is more readable now and works correct. Opinions are welcome. best wishes. 
AP's
Also I have refreshed the APrecord site:
News: More compact and all AP24,AP25,AP26,AP27 are addapted from PrimeGride under largest known APk > Record History [url]http://www.pzktupel.de/JensKruseAndersen/aprecords.htm[/url] 
PI_10,PI_11
Update:
PI_10(10^20) and PI_11(10^20) are known. There are 10,230,073 prime 10tuplets and 411,551 prime 11tuplets up to 10^20. Next: PI_12(10^21) Estimated time: 13d 
PI_12(X)
PI_12(10^21) is abort, files are lost...no passion to repeat it. :no:

PI_3(10^17) is known !
Many thanks to Peter Kaiser for calculating this PItable !
1e16 stepspattern 1pattern 2 01, 624.026.299.748, 624.025.508.307 12, 554.086.126.694, 554.084.938.742 23, 531.304.381.448, 531.303.963.615 34, 517.196.606.698, 517.196.768.303 45, 507.032.665.908, 507.031.620.118 56, 499.125.675.906, 499.125.938.384 67, 492.675.249.232, 492.676.420.737 78, 487.241.567.509, 487.241.788.030 89, 482.557.480.218, 482.557.274.458 910, 478.443.472.603, 478.444.163.939 [B]total: 5.173.689.525.964, 5.173.688.384.633 [/B] 
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