Counting up in prime numbers #2

RogerE · Post by **RogerE** » 15 Jun 2020 04:30

More than 14hrs later:

113,891

BlackTuesday · Post by **BlackTuesday** » 15 Jun 2020 07:34

113,899
113,903

@RogerE, thanks for the link to that thread

RogerE · Post by **RogerE** » 16 Jun 2020 01:07

113,909

__________________________________________________________
Some thoughts about BlackTuesday's previous post.

Yes, BlackTuesday, two primes in a run of six numbers should be regarded as a "dense as possible" configuration; three primes in a run of eight numbers should also be so regarded, and so on.

Some theoretical definitions

• Fixed number of primes: For any positive integer k, let L(k) be the minimum number such that there is a run of L(k) consecutive integers which contains k primes, all of which are greater than the first k primes. All examples of such a run of L(k) numbers must begin and end with a prime: if p is the first member of the run, then the last member of the run is q = p + L(k) – 1, and L(k) = q – p + 1.

Then L(k) is always odd, and the first few values are
L(1) = 1 [e.g. {3}];
L(2) = 3 [e.g. {5, 7}];
L(3) = 7 [e.g. {7, 11, 13}];
L(4) = 9 [e.g. {11, 13, 17, 19}];
L(5) = 13 [e.g. {101, 103, 107, 109, 113}] etc.
The conjecture is that for each k there are infinitely many examples (known to be true for k = 1, but not known to be true for any k ≥ 2).

• Fixed run size: For any positive integer n, let M(n) be the maximum number of primes, all greater than n, which occur in a run of n consecutive integers. An example of any such run of n numbers containing M(n) primes is a neighbour configuration if the run begins and ends with a prime: if p is the first member of the run, then the last member of the run is q = p + n – 1, and n = q – p + 1. A neighbour configuration is maximal if it is not contained in a larger neighbour configuration.

If M(n) > M(n-1) then n is odd; if n is odd then M(n+1) = M(n). A maximal configuration can only occur when n is odd.
M(1) = 1 [e.g. {2}];
M(2) = 1 [e.g. {3}];
M(3) = M(4) = 2 [e.g. {5, 7}];
M(5) = M(6) = 2 [e.g. {7, 11}];
M(7) = M(8) = 3 [e.g. {11, 13, 17}];
M(9) = M(10) = 4 [e.g. {11, 13, 17, 19}, {101, 103, 107, 109}];
M(11) = M(12) = 4 [e.g. {37, 41, 43, 37}];
M(13) = M(14) = 5 [e.g. {97, 101, 103, 107, 109}];
M(15) = M(16) = 5 [e.g. {17, 19, 23, 29, 31}];
M(17) = M(18) = 6 [e.g. {97, 101, 103, 107, 109, 113}] etc.

The case represented by 113,899 and 113,903 is maximal neighbour configuration for M(5) = 2.

/RogerE

BlackTuesday · Post by **BlackTuesday** » 16 Jun 2020 14:31

113,921

Thanks a lot RogerE for your detailed explanation

RogerE · Post by **RogerE** » 17 Jun 2020 03:44

BlackTuesday wrote: ↑
16 Jun 2020 14:31
Thanks a lot RogerE for your detailed explanation

You're welcome

I hope that hatter doesn't mind my contribution — he started this thread, so he retains authorship entitlement to comment, if he wishes, on what gets posted ...

113,933

RogerE · Post by **RogerE** » 18 Jun 2020 03:54

After 24 hrs elapsed tome:

113,947

RogerE · Post by **RogerE** » 18 Jun 2020 16:05

After 12hrs:

113,957

Next two come soon, but not quite close enough to count as a "configuration".

RogerE

RogerE · Post by **RogerE** » 19 Jun 2020 04:05

After another 12hrs:

113,963

BlackTuesday · Post by **BlackTuesday** » 19 Jun 2020 07:51

113,969

RogerE · Post by **RogerE** » 19 Jun 2020 18:45

113,983

BlackTuesday · Post by **BlackTuesday** » 20 Jun 2020 06:13

113,989

RogerE · Post by **RogerE** » 20 Jun 2020 14:20

Breakthrough!

114,001

RogerE · Post by **RogerE** » 21 Jun 2020 04:01

After almost 14hrs:

114,013

BlackTuesday · Post by **BlackTuesday** » 22 Jun 2020 00:39

114,031

RogerE · Post by **RogerE** » 22 Jun 2020 23:20

114,041

114,043

Gemini — twin happiness

RogerE · Post by **RogerE** » 23 Jun 2020 14:08

After about 15hrs:

114,067

RogerE · Post by **RogerE** » 24 Jun 2020 22:31

After more than 30hrs!

114,073

114,077

— another "as dense as possible" group in a run of five consecutive numbers...

RogerE · Post by **RogerE** » 26 Jun 2020 01:28

After another wait of more than 24hrs (where are you all?):

114,083

RogerE · Post by **RogerE** » 27 Jun 2020 23:42

Ages later, still no other contribution, so here's the next:

114,089

RogerE · Post by **RogerE** » 30 Jun 2020 03:59

More than 2 days later — where are the other posters for this thread?

114,113

/RogerE

RogerE · Post by **RogerE** » 01 Jul 2020 01:51

23hrs later

114,143

Stamp collector · Post by **Stamp collector** » 01 Jul 2020 03:34

114157
114161
114167

RogerE · Post by **RogerE** » 01 Jul 2020 06:46

Hmmm, 114,157 and 114,161 together are a configuration (2 primes in a run of five is maximal).
But 114,167 is not part of that configuration (3 primes in a run of eleven is not maximal
— for example 97, 101, 103, 107 are four primes in a run of eleven.
(An even denser occurrence of four primes is 101, 103, 107, 109 so 4 is already the best possible
in runs of nine.)
So, 114,167 should have been the next post.

After that we have:
114,193
114,197
114,199
114,203
This is a maximal configuration (4 primes in a run of eleven)

/RogerE

Stamp collector · Post by **Stamp collector** » 03 Jul 2020 05:06

114,217
114,221

BlackTuesday · Post by **BlackTuesday** » 04 Jul 2020 02:10

114,229

I'm back

RogerE · Post by **RogerE** » 04 Jul 2020 02:48

Nice to see other regular participants back on this thread

114,259

BlackTuesday · Post by **BlackTuesday** » 04 Jul 2020 10:57

114,269

RogerE · Post by **RogerE** » 04 Jul 2020 14:12

114,277

114,281

Two primes in a run of five integers: maximal configuration.

BlackTuesday · Post by **BlackTuesday** » 05 Jul 2020 03:36

114,299

RogerE · Post by **RogerE** » 05 Jul 2020 11:16

114,311

BlackTuesday · Post by **BlackTuesday** » 05 Jul 2020 18:23

114,319

RogerE · Post by **RogerE** » 05 Jul 2020 22:53

114,329

BlackTuesday · Post by **BlackTuesday** » 06 Jul 2020 08:30

114,343

RogerE · Post by **RogerE** » 06 Jul 2020 16:55

114,371

BlackTuesday · Post by **BlackTuesday** » 08 Jul 2020 04:37

114,377

RogerE · Post by **RogerE** » 08 Jul 2020 16:30

114,407

Of course, 114411 and 114477 will be composite, because they "must be" multiples of 11
— the alternating sum/difference of their digits is 0.
In fact any number greater than 11 with digits occurring in repeated pairs is a multiple of 11.
There are some posts about this and related ideas. The main one for 11 is
https://www.stampboards.com/viewtopic.php?f=6&t=40&start=127

/RogerE

BlackTuesday · Post by **BlackTuesday** » 11 Jul 2020 19:43

114,419

RogerE · Post by **RogerE** » 12 Jul 2020 04:14

114,451

BlackTuesday · Post by **BlackTuesday** » 12 Jul 2020 10:09

114,467

RogerE · Post by **RogerE** » 12 Jul 2020 13:00

114,473

BlackTuesday · Post by **BlackTuesday** » 12 Jul 2020 19:17

114,479

RogerE · Post by **RogerE** » 13 Jul 2020 01:06

114,487

BlackTuesday · Post by **BlackTuesday** » 13 Jul 2020 09:54

114,493

RogerE · Post by **RogerE** » 13 Jul 2020 10:04

114,547

turtle-bienhoa · Post by **turtle-bienhoa** » 13 Jul 2020 13:39

114553

RogerE · Post by **RogerE** » 13 Jul 2020 16:17

114,571

BlackTuesday · Post by **BlackTuesday** » 13 Jul 2020 21:22

114,577

RogerE · Post by **RogerE** » 13 Jul 2020 22:30

114,593

BlackTuesday · Post by **BlackTuesday** » 14 Jul 2020 21:18

114599

114601

Two primes in a run of three integers

RogerE · Post by **RogerE** » 15 Jul 2020 12:25

Recall that two primes in a run of three integers (>2) are twin primes.

Not so close, but still maximum possible, are two primes in a run of five integers (>3)
since three consecutive odd integers must include a multiple of 3:

114,613

114,617

/RogerE