QUADRUPLET PRIMES AND THEIR GAPS - Page 10

Cybertronic · 2022-11-19, 08:46

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

Continue with triplets https://www.mersenneforum.org/showthread.php?t=28223

Bobby Jacobs · 2022-11-24, 21:20

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.

chalsall · 2022-11-24, 21:28

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.

Cybertronic · 2022-12-13, 08:56

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

mart_r · 2023-01-02, 22:51

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).

mart_r · 2023-01-09, 19:43

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.

Cybertronic · 2023-01-10, 09:47

Under
https://pzktupel.de/RecordGaps/GAP04.php
you get at end of the page the current status as zip-file

mart_r · 2023-01-26, 18:57

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;


fn="quadgaps_output.txt"; \\ filename for output

\\ 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;
	print("quads coprime to "o"#: "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
)
}

Cybertronic · 2023-01-26, 20:02

Thanks Martin !
" The First Occurrence Gap Between Prime Quadruplets" - list up to 2.08e16 now...
https://pzktupel.de/RecordGaps/GAP04.php

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, 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

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
GAPS BETWEEN PRIME PAIRS (Twin Primes)	rudy235	Prime Gap Searches	262	2023-01-24 19:09
Gaps between maximal prime gaps	Bobby Jacobs	Prime Gap Searches	52	2020-08-22 15:20
Prime Quadruplet Emirps	Trejack	Puzzles	11	2016-04-22 19:27
Gaps of Primes?	PawnProver44	Miscellaneous Math	10	2016-04-10 19:32
Top-5 prime quadruplet found	gd_barnes	Riesel Prime Search	10	2007-10-27 01:54