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2020-01-04, 01:33   #12
rudy235

Jun 2015
Vallejo, CA/.

2×33×19 Posts

Quote:
 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?

2020-01-04, 13:35   #13
mart_r

Dec 2008
you know...around...

2·73 Posts

Quote:
 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.

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

2020-01-04, 14:26   #14
robert44444uk

Jun 2003
Oxford, UK

22×32×5×11 Posts

Quote:
 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

Last fiddled with by robert44444uk on 2020-01-04 at 14:26

2020-01-05, 04:30   #15
rudy235

Jun 2015
Vallejo, CA/.

100000000102 Posts

Quote:
 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

 2020-01-05, 08:20 #16 robert44444uk     Jun 2003 Oxford, UK 22×32×5×11 Posts 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.
2020-01-06, 08:34   #17
Thomas11

Feb 2003

191510 Posts

Quote:
 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:
Code:
perl update_gaplist.pl logfile.txt
This will add all new first instances to gaplist.txt

Code:
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:
Code:
21764 (1) -- unknown
22700 (6) -- unknown
22807 (1) -- unknown
23143 (1) -- unknown
24100 (6) -- unknown
The values in parentheses are "gap mod 7".
Attached Files

Last fiddled with by Thomas11 on 2020-01-06 at 08:37

2020-01-06, 09:55   #18
robert44444uk

Jun 2003
Oxford, UK

22·32·5·11 Posts

Quote:
 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: Code: perl update_gaplist.pl logfile.txt This will add all new first instances to gaplist.txt Code: 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: Code: 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

2020-01-06, 19:52   #19
mart_r

Dec 2008
you know...around...

2AE16 Posts

Quote:
 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.

2020-01-07, 04:45   #20
rudy235

Jun 2015
Vallejo, CA/.

2×33×19 Posts

Quote:
 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.

2020-01-07, 17:09   #21
robert44444uk

Jun 2003
Oxford, UK

7BC16 Posts

Quote:
 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:

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

Last fiddled with by robert44444uk on 2020-01-07 at 17:09

2020-01-07, 18:37   #22
mart_r

Dec 2008
you know...around...

12568 Posts

Quote:
 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

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