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2022-11-19, 08:46   #100
Cybertronic

Jan 2007
Germany

11528 Posts
10^16 complete

Searching up to 10^16 now complete. No new gap record, but one near missing. (k=322493, 9911752981863911)

Attached Files
 gaplist1e16.zip (1.42 MB, 11 views)

Last fiddled with by Cybertronic on 2022-11-19 at 08:46

2022-11-24, 21:20   #101
Bobby Jacobs

May 2018

23×5×7 Posts

Quote:
 Originally Posted by rudy235 Here is a list of first instances For the Quadruplets (notice how different it is from the list of First Differences between Twin Primes) But first a review GAPS between primes P≥3 Minumum gasp is 2 Maximum gap Until Dec 2019 is the gap of 5,103,138 after a PRP of 216,849 digits. GAP between Twins p,p+2 to q,q+2 these gaps between p and q are always of the form 6k Minimum gap (expressed as k) 1 Maximum gap yet found 518546 As to the gap between Quadruplet primes we define the number k as the distance between the first element of the QP (or alternatively of the barycenter) divided by 30 The barycenter –or Pivot– is the central number (it ends in 5) For instance the baricenter of the quadruplet {821, 823, 827, 829 is 825 A159910 FIRST OCCURRENCE Code: Gap After p= 1 1006301 rudy235 2 NONE 3 11 rudy235 4 1022381 rudy235 5 NONE 6 3512051 rudy235 7 1871 rudy235 8 632081 rudy235 9 NONE 10 71121831 rudy235 11 15731 rudy235 12 NONE 13 1481 rudy235 14 1155611 rudy235 15 1068251 rudy235 16 NONE 17 18911 rudy235 18 284741 rudy235 19 NONE 20 12390011 rudy235 21 191 rudy235 22 821 rudy235 23 NONE 24 3837131 rudy235 25 875261 rudy235 26 NONE 27 854921 rudy235 28 10865291 rudy235 29 18041 rudy235 30 NONE 31 958541 rudy235 32 680291 rudy235 33 NONE 34 299471 rudy235 35 1063961 rudy235 36 3995441 rudy235 37 NONE 38 4404551 39 5733525 40 NONE The largest gap I have found is a gap of 1897 between the Quadruplet of barycenter 3,741,165 and that of 3,798,075 3'798,075-3,741,165= 56910 . 56,910/30 = 1897
You should remove the 7 from the gap of 10. It is at 1121831 instead of 71121831. Also, 5733525 is not prime. The gap of 39 is at 2081.

2022-11-24, 21:28   #102
chalsall
If I May

"Chris Halsall"
Sep 2002

22·5·7·79 Posts

Quote:
 Originally Posted by Bobby Jacobs You should remove the 7 from the gap of 10. It is at 1121831 instead of 71121831. Also, 5733525 is not prime. The gap of 39 is at 2081.
Noise.

2022-12-13, 08:56   #103
Cybertronic

Jan 2007
Germany

2·3·103 Posts
Status 13th Dec 2022

Up to 1.03e16 ( and double check ) done.
See zip-file for gaps and open gaps
Attached Files

Last fiddled with by Cybertronic on 2022-12-13 at 09:15

2023-01-02, 22:51   #104
mart_r

Dec 2008
you know...around...

24×53 Posts
An update

New first occurrence gaps between prime quadruplets, 1030e13 to 1278e13, see attachment.
One new maximal gap: 334884 (*30) - see also https://pzktupel.de/RecordGaps/GAP04.php (Thanks Norman! )
Smallest first occurrence not yet found: 171032 (*30).
Attached Files

2023-01-09, 19:43   #105
mart_r

Dec 2008
you know...around...

24×53 Posts
Statistics that (almost surely) noone had considered yet

Quote:
Originally Posted by Cybertronic
A just for fun table....
Quote:
 gaps found below record gap / record gap 4162 / 5217 =>79.77% 4251 / 5247 =>81,0% 4576 / 5252 => 87,1% 4583 / 5320 => 86,14% 4589 / 5507 => 83,33% 4742 / 5782 => 82,01% 4878 / 5940 => 82,12% 4989 / 5940 => 83,8% <- status now Next: record gap ~6100 ?
A similar table for gaps between prime quadruplets:
Code:
Column A: number of gaps found <= maximal gap
Column B: number of all admissible gaps <= maximal gap
Column C: ratio
Taken at all points < 1e16 where a new maximal gap was found.

A       B       C
1       2  50.00%
2      15  13.33%
3      16  18.75%
6      28  21.43%
7      52  13.46%
8      90   8.89%
18     153  11.76%
20     213   9.39%
32     215  14.88%
38     316  12.03%
44     392  11.22%
61     575  10.61%
105     686  15.31%
156     705  22.13%
158     952  16.60%
235    1347  17.45%
263    1355  19.41%
309    1705  18.12%
327    1987  16.46%
419    2251  18.61%
588    2725  21.58%
658    3758  17.51%
956    3787  25.24%
1125    4076  27.60%
1185    4223  28.06%
1258    4536  27.73%
1472    4712  31.24%
1533    6382  24.02%
1846    7520  24.55%
2882    9417  30.60%
4082   10363  39.39%
4353   10498  41.47%
4363   12228  35.68%
4692   12762  36.77%
4886   12841  38.05%
4944   13270  37.26%
5243   15122  34.67%
6093   20144  30.25%
9614   20965  45.86%
9798   21768  45.01%
10583   21973  48.16%
11953   23909  49.99%
12156   25488  47.69%
12698   25891  49.04%
12906   26032  49.58%
13403   28530  46.98%
13762   31820  43.25%
15568   31821  48.92%
15906   32106  49.54%
16058   32625  49.22%
16992   35713  47.58%
17904   36920  48.49%
19148   60732  31.53%
29070   60995  47.66%
34375   69427  49.51%
36178   69620  51.96%
38463   70373  54.66%
40129   81488  49.25%
45330   82303  55.08%
46326   92492  50.09%
52329  100675  51.98%
56087  108046  51.91%
57173  109605  52.16%
58840  116307  50.59%
65192  139317  46.79%
86077  154755  55.62%
88233  155827  56.62%
98733  156395  63.13%
99917  160722  62.17%
100516  166279  60.45%
104025  187737  55.41%
111457  193135  57.71%
118534  194895  60.82%
123449  199898  61.76%
124321  210599  59.03%
131102  215472  60.84%
132495  222533  59.54%
137907  234190  58.89%
Fun fact: the (conjecturally) only first occurrence gaps that were at some point the smallest not yet found which were not congruent to 2 or 5 mod 7 are:
[120, 215370, 392820, 979320, 1561440].
Currently the 27th smallest gap not yet found as first occurrence is the first that's not 2 or 5 mod 7, corroborating that the above list of five numbers may be finite.

 2023-01-10, 09:47 #106 Cybertronic     Jan 2007 Germany 26A16 Posts Gaps between prime quadruplets , zip-file Under https://pzktupel.de/RecordGaps/GAP04.php you get at end of the page the current status as zip-file Last fiddled with by Cybertronic on 2023-01-10 at 09:55
2023-01-26, 18:57   #107
mart_r

Dec 2008
you know...around...

35016 Posts
Memento numeri

Results 1.278e16 to 2.08e16 are attached.
Smallest first occurrence not yet found: 184584 (*30).

If a volunteer or two could join in, the search may be carried out to 1e17 within a week or two. In any case, I'll reserve up to 2.4e16 for now.
This Pari program reaches a speed of more than 2e9 per second per core on my more than seven years old PC with an Intel i7-4790 @ 3.6 GHz, I'd guess more than twice the speed is possible on a more modern machine. Just enter interval start (k) and interval end (t) at the beginning. (Note that for test runs with smaller numbers, not all gaps may be found due to the "jump" parameters (j,u) below; these may also be altered if one feels comfortable enough with that.)
Code:
{
\\ interval start, interval end:
k=24000*10^12;
t=24100*10^12;

\\ parameters that may be tweaked as search progresses:
g=184580*30; \\ min gap size for output
j=14080;     \\ jump #offsets
u=3200;      \\ check #offsets before backtracking gap
\\ (j+u)*r/z - particularly the largest difference between (j+u) offsets - must be smaller than g !

\\ compute offsets mod 23#
gettime();
r=223092870;
z=700245;
v=vector(z);w=v;w[1]=11;a=1;n=30;o=5;
while(n<r,
o=nextprime(o+1);
l=0;
for(m=0,o-1,
for(b=1,a,
if(gcd(n*m+w[b],n*o)==1&&gcd(n*m+w[b]+2,n*o)==1&&gcd(n*m+w[b]+6,n*o)==1&&gcd(n*m+w[b]+8,n*o)==1,l++;v[l]=n*m+w[b])
)
);
w=v;
n*=o;
a=l;
);
if(l==z&&n==r,print("OK, "gettime()" ms"),print("Error, check parameters r,z!");break());

c=vector(131072); \\ array for tracking which gaps have been found
s=1;forprime(y=29,3089,s*=y);    \\ for 1st gcd check
x=1;forprime(y=3109,11399,x*=y); \\ for 2nd gcd check
print("searching, k="k);
k=r*floor(k/r);
\\ find first quadruplet > k
a=1;
i=1;
while(i,
p=k+v[a];
a++;
if(a>z,a=1;k+=r);
if(gcd(p,s)==1,if(gcd(p+2,s)==1,if(gcd(p+6,s)==1,if(gcd(p+8,s)==1,
if(gcd(p,x)==1,if(gcd(p+2,x)==1,if(gcd(p+6,x)==1,if(gcd(p+8,x)==1,
if(ispseudoprime(p),if(ispseudoprime(p+2),if(ispseudoprime(p+6),if(ispseudoprime(p+8),i=0))))
))))
))))
);
\\ find quadruplet gaps > g in range (k,t), output every gap size once (but all gaps > #c*30+g)
while(k<t,
h=1;
while(h,
a+=j;
if(a>z,a-=z;k+=r);
b=0;
o=k;d=a;
i=1;
while(i,
p=k+v[a];
a++;
b++;
if(a>z,a=1;k+=r);
if(gcd(p,s)==1,if(gcd(p+2,s)==1,if(gcd(p+6,s)==1,if(gcd(p+8,s)==1,
if(gcd(p,x)==1,if(gcd(p+2,x)==1,if(gcd(p+6,x)==1,if(gcd(p+8,x)==1,
if(ispseudoprime(p),if(ispseudoprime(p+2),if(ispseudoprime(p+6),if(ispseudoprime(p+8),i=0))))
))))
))))
);
if(b>u,h=0)
);
q=p;
e=k;f=a;
i=1;
k=o;a=d;
while(i,
a-=1;
if(a<1,a=z;k-=r);
p=k+v[a];
if(gcd(p,s)==1,if(gcd(p+2,s)==1,if(gcd(p+6,s)==1,if(gcd(p+8,s)==1,
if(gcd(p,x)==1,if(gcd(p+2,x)==1,if(gcd(p+6,x)==1,if(gcd(p+8,x)==1,
if(ispseudoprime(p),if(ispseudoprime(p+2),if(ispseudoprime(p+6),if(ispseudoprime(p+8),i=0))))
))))
))))
);
if(q-p>g,
n=(q-p-g)/30;
if(n>#c,n=#c);
if(!c[n],
print((q-p)/30",  "p"  ("floor(gettime()/1000)" s)");
write(fn,(q-p)/30",  "p);
if(n<#c,c[n]=1)
)
);
k=e;a=f
)
}
Attached Files

 2023-01-26, 20:02 #108 Cybertronic     Jan 2007 Germany 11528 Posts Thanks Martin ! " The First Occurrence Gap Between Prime Quadruplets" - list up to 2.08e16 now... https://pzktupel.de/RecordGaps/GAP04.php Last fiddled with by Cybertronic on 2023-01-26 at 20:03

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