QUADRUPLET PRIMES AND THEIR GAPS

[rudy235]

Originally Posted by Thomas11

Here are the first occurrences of gaps between quadruplets searched up to 1e12.

Largest gap: 85025 342611795411

Smallest unknown gap: 21764

What are the next 7 unknown gaps? And how many of those are either 1 mod 7 or 6 mod 7?

[mart_r]

Originally Posted by rudy235

What are the next 7 unknown gaps? And how many of those are either 1 mod 7 or 6 mod 7?

Pencil, paper and some simple modular arithmetic suffice to show that there's a high probability that all of them are +/-1 mod 7.

gap/30 mod 7  number of open residues
2, 5          0 (as you already noticed)
1, 6          1
3, 4          2
0             3

[robert44444uk]

Originally Posted by rudy235

What are the next 7 unknown gaps? And how many of those are either 1 mod 7 or 6 mod 7?

At 2e12 looks like the first is 25964 and the next is 26074

Has anyone got a perl or an exe file that can take the results from Thomas11's program and make a list of first instances?

I'll take this to 1e13

[rudy235]

Originally Posted by robert44444uk

At 2e12 looks like the first is 25964 and the next is 26074

Has anyone got a perl or an exe file that can take the results from Thomas11's program and make a list of first instances?

I'll take this to 1e13

...and while we are asking for thigs, it would be useful to have the still unknown gaps.
Because this search is being done exhaustively they should not be any C?C, just gaps not yet discovered like:25964 and 26074

[robert44444uk]

At 2e13 the earliest I could find without a value appears to be 42981

The largest was 129489 16837633318811

These results should be considered provisional until I get a program to handle the results.

[Thomas11]

Originally Posted by robert44444uk

Has anyone got a perl or an exe file that can take the results from Thomas11's program and make a list of first instances?

Here are two perl scripts for managing the gap results.

Both assume that your master gap list is stored in the file "gaplist.txt" (an example file with gaps up to 1e12 is included).

Usage is as follows:

perl update_gaplist.pl logfile.txt

This will add all new first instances to gaplist.txt

perl check_gaplist.pl

This is a quick check for any wrong gaps and reports the unknown gaps.

The screen output of the latter will look as the following:

21764 (1) -- unknown
22700 (6) -- unknown
22807 (1) -- unknown
23143 (1) -- unknown
24100 (6) -- unknown

The values in parentheses are "gap mod 7".

[robert44444uk]

Originally Posted by Thomas11

Here are two perl scripts for managing the gap results.

Both assume that your master gap list is stored in the file "gaplist.txt" (an example file with gaps up to 1e12 is included).

Usage is as follows:

perl update_gaplist.pl logfile.txt

This will add all new first instances to gaplist.txt

perl check_gaplist.pl

This is a quick check for any wrong gaps and reports the unknown gaps.

The screen output of the latter will look as the following:

21764 (1) -- unknown
22700 (6) -- unknown
22807 (1) -- unknown
23143 (1) -- unknown
24100 (6) -- unknown

The values in parentheses are "gap mod 7".

Good man, Thomas11!

I'll post the results when I get to 1e14 - at the moment I am at 3e13 with 52048 as the smallest unknown.

The largest gaps to 3e13 are:

111504 29595233745191
112105 26248083591881
113152 28184538475271
113467 25801593675041
114083 8547351574961
115224 15114108020021
118307 18317170605491
125692 18888105484811
129489 16837633318811

[mart_r]

Originally Posted by rudy235

What are the next 7 unknown gaps? And how many of those are either 1 mod 7 or 6 mod 7?

The really interesting question would be "is the number of first holes that are not 1 or 6 mod 7 finite?"

You might have a look at similar data from prime gaps (check successive maximal values of p in increasing gaps g and how many are g mod 3 = 1 or 2) or twin prime gaps (how many g/6 mod 5 = 1 or 4). I'd guess the bias for quad prime gaps should be even stronger.

[rudy235]

Originally Posted by mart_r

The really interesting question would be "is the number of first holes that are not 1 or 6 mod 7 finite?"

You might have a look at similar data from prime gaps (check successive maximal values of p in increasing gaps g and how many are g mod 3 = 1 or 2) or twin prime gaps (how many g/6 mod 5 = 1 or 4). I'd guess the bias for quad prime gaps should be even stronger.

I believe that all gaps ≤ N (except the aforementioned with 2 or 5 mod7) are to be found if you search high enough. I don't quite understand what is meant by first holes.

[robert44444uk]

Originally Posted by robert44444uk

Good man, Thomas11!

I'll post the results when I get to 1e14 - at the moment I am at 3e13 with 52048 as the smallest unknown.

The largest gaps to 3e13 are:

….
129489 16837633318811

Now at 9e13, and 52048 is still the smallest unknown. Large gaps were:

140945 30709975578251
143014 68802225596051
143374 73374129557621
148435 48935654916221
151264 43785651890171
153447 47998980412211
159414 74498625237131
162830 55341128536691

[mart_r]

Originally Posted by rudy235

I believe that all gaps ≤ N (except the aforementioned with 2 or 5 mod7) are to be found if you search high enough. I don't quite understand what is meant by first holes.

First holes, in this case, are the gaps for which you have to search farthest to find them.

For ordinary prime gaps, this is A100180