new prime k-tuplet page

[Batalov]

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 6-Tuplet there (currently it is not 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 all 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. I liked CPAPs because they are hard. Easy things are boring.)

[Cybertronic]

You mean, the SP-Table over all is okay, but I should extened a Top-10(20) list for each SP-kind ,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 k-tuplet page have now all kinds and was refreshed yesterday.
https://www.pzktupel.de/ktuplets

[Cybertronic]

I have refreshed Jens K Andersens " Simultaneous Primes Prime Page"


http://www.pzktupel.de/JensKruseAnde...imultprime.htm


I hope it is more readable now and works correct. Opinions are welcome.
best wishes.

[Cybertronic]

Also I have refreshed the AP-record site:


News: More compact and all AP24,AP25,AP26,AP27 are addapted from PrimeGride under largest known AP-k -> Record History



http://www.pzktupel.de/JensKruseAndersen/aprecords.htm

[Cybertronic]

Update:
PI_10(10^20) and PI_11(10^20) are known.


There are 10,230,073 prime 10-tuplets and 411,551 prime 11-tuplets up to 10^20.


Next: PI_12(10^21)
Estimated time: 13d

[Cybertronic]

PI_12(10^21) is abort, files are lost...no passion to repeat it.

[Cybertronic]

Many thanks to Peter Kaiser for calculating this PI-table !


1e16 steps|pattern 1|pattern 2



0-1, 624.026.299.748, 624.025.508.307
1-2, 554.086.126.694, 554.084.938.742
2-3, 531.304.381.448, 531.303.963.615
3-4, 517.196.606.698, 517.196.768.303
4-5, 507.032.665.908, 507.031.620.118
5-6, 499.125.675.906, 499.125.938.384
6-7, 492.675.249.232, 492.676.420.737
7-8, 487.241.567.509, 487.241.788.030
8-9, 482.557.480.218, 482.557.274.458
9-10, 478.443.472.603, 478.444.163.939

total: 5.173.689.525.964, 5.173.688.384.633

[Cybertronic]

is new on the page.


I: https://www.pzktupel.de/APTuple.html

II: https://www.pzktupel.de/ConsTupleSM.html


greetings


Norman

[Cybertronic]

On prime k-tuplet mainpage and smallest x000 digit page I have now added for more than 1000 digits the "Primo-Certificates" under [click].



https://pzktupel.de/ktuplets.php
https://pzktupel.de/smx000.html


best


N.

[MattcAnderson]

Hi all,

For the experts familiar with Hardy-Littlewood 2nd conjecture (not compatible with k-tuple conjecture <in my humble opinion> IMHO {k-tuple conjecture is probably true})

Let prime counting function pi(x) be the count of the number of primes less than or equal to x.
See references
Thomas J Englesma k-tuples.html


Also, Norman (Cybertronic at mersenneforum.org) has a very extensive database
See largest.php

Note that there is a -still- unknown 19-tuple and 22-tuples

to be specific , the 19-tuple has pattern
d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66, 70, 76
So the difference between the largest and the smallest prime number in this set is 76.

If we calculate more and more k-tuples, then maybe some day we can have a 446-tuple,
and we can be right on the edge of this counterexample.

As Tarence Tao said in one of his lectures, "I do not believe that there is a conspiracy in the numbers."
He was talking about the k-tuple conjecture (First Hardy-Littlewood Conjecture)

It has been proven maybe ~1970 that k-tuple conjecture and this other one are not both true.

So we let time pass, and Norman's database continues to grow, and someday we know
- even bigger numbers -

That is all for now.

Have a nice day.

Matthew

This 447-tuple for someday my be thousands of years from now, at the rate that we are gong.

<Song> "In the year 2525"
https://www.youtube.com/watch?v=izQB2-Kmiic
</song>

TLDR (Too Long Didn't Read) Hey Guy! wouldn't it be great if we also calculated this other thing? That would be soooooo cool.

[JeppeSN]

Originally Posted by MattcAnderson

It has been proven maybe ~1970 that k-tuple conjecture and this other one are not both true.

For people who know and love some mathematics, but are not specialists, there is an easy-to-read paper from 1974 about how they discovered this: Ian Richards, On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem, https://doi.org/10.1090/S0002-9904-1974-13434-8 (free access to full paper).

At the time they knew the "width" of a minimal example tuple would be more than 500 but less than 20'000. The 447-tuple has width 3159. To quote Wikipedia, "For example, an admissible k-tuple (or prime constellation) of 447 primes can be found in an interval of y = 3159 integers, while π(3159) = 446."

/JeppeSN