20220128, 17:32  #23 
Jun 2003
Suva, Fiji
2^{3}·3·5·17 Posts 
I thought I would look at this again, using mart_r's table. The idea is that someone could create an entry to OEIS.
These are the first instance primes where the number of twins shown (of which the first instance prime was the smaller of a twin) are clustered in the narrowest range of integers. Excluded are nonadmissible patterns. The list confirms the smallest 6 found by henryzz and lists the Jens Kruse Andersen 8 twin discovery. #of twins First prime Twin pattern OEIS 3 179 {0,2,12,14,18,20} A253627 4 626597 {0,2,12,14,24,26,30,32} A253624 5 39713433671 {0,2,6,8,18,20,30,32,36,38} A256842 6 1256522812841 {0,2,6,8,18,20,30,32,36,38,48,50] Found in this group https://www.mersenneforum.org/showpo...94&postcount=5 7 1135141716537971 {0,2,6,8,18,20,30,32,36,38,48,50,60,62} Listed at http://www.balcro.com/fix5tp39.html as part of the search for 5twins back in the day. 8 17625750738756291797 {0,2,12,14,24,26,30,32,42,44,54,56,72,74,84,86} Listed at https://primes.utm.edu/curios/page.p...756291797.html I'm looking at all patterns, and have reserved 9 and 10 twins Last fiddled with by robert44444uk on 20220128 at 17:52 
20220205, 18:37  #24 
May 2018
281 Posts 
The smallest admissible distance between 2 consecutive primes is 2. These are twin primes (p, p+2). The smallest admissible distance between 2 consecutive pairs of twin primes is 6. These are twin twin primes, or prime quadruplets (p, p+2, p+6, p+8). The smallest admissible distance between 2 consecutive sets of twin twin primes is 30. These are twin twin twin primes, or twin prime quadruplets (p, p+2, p+6, p+8, p+30, p+32, p+36, p+38). The smallest set of twin twin twin primes is 1006301, 1006303, 1006307, 1006309, 1006331, 1006333, 1006337, 1006339. What is the smallest admissible distance between 2 consecutive sets of twin twin twin primes? What is the smallest set of twin twin twin twin primes?

20220206, 06:18  #25 
Jun 2003
Suva, Fiji
2^{3}·3·5·17 Posts 
I'm up to 6e19 in the search for 9 twins (using all 4 patterns) in 105 digits without a result.

20220206, 11:06  #26  
Dec 2008
you know...around...
853 Posts 
Quote:
Playing the game further, the smallest admissible distance between two of those is 2310. You probably won't live long enough for witnessing the discovery of the first occurrence. Last fiddled with by mart_r on 20220206 at 11:08 

20220206, 18:05  #27 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{3}×293 Posts 
3# = 6
5# = 30 7# = 210 11# = 2310 Is the 7# distance impossible, or just that none is found so far? ETA: the pattern can be extended towards the lower side for 3, 5, 7 with 2# (p, p+2, p+2, p+4) ETA II: “half” twin primes are very finite. Twin primes are very much probably infinite. Is it known if TwinTwin primes are finite? (I assume the knowledge to their infinity would definitely be unknown since even infinity of twins is unknown). Last fiddled with by a1call on 20220206 at 18:40 
20220206, 19:14  #28  
Feb 2017
Nowhere
13·479 Posts 
Quote:
Code:
? v=[0, 2, 6, 8, 30, 32, 36, 38, 420, 422, 426, 428, 450, 452, 456, 458];w=vector(#v,i,x+ v[i]);w2=vector(#v, i,x + v[i]+210);f=prod(i=1,#v,w[i]*w2[i]);forprime(p=2,32,fp=Mod(1,p)*f;if(Mod(fp,x^px)==0,print(p))) 11 19 

20220206, 20:04  #29  
Einyen
Dec 2003
Denmark
19×181 Posts 
Quote:
11281963036964038421 + [80,122,318,330,332,402] Is it theoretically possible with only those 16 primes between p and p+458 ? Edit: Yes, it is possible I found out myself, but probably much harder to find an example. Last fiddled with by ATH on 20220206 at 20:54 

20220206, 21:48  #30  
Dec 2008
you know...around...
853 Posts 
Quote:
The currently largest known 16tuplet has 35 digits, log is 78.56. The gap between the two double quadruplets is 382, which equals a merit of 4.86 in that region. (Also don't forget the possibility of up to two primes between the quadruplets.) Finding such a pattern without any prime inbetween will be very hard, at least for now. At the least, it might be interesting to figure out the tradeoff between number size and theoretical number of 16tuplets to be found until the gap appears, for a possible future computation on a quantum chip. 

20220206, 22:24  #31 
Einyen
Dec 2003
Denmark
19·181 Posts 
Even the next one is possible with no primes in between with only those 32 primes between p and p+2768.
p + [0, 2, 6, 8, 30, 32, 36, 38, 420, 422, 426, 428, 450, 452, 456, 458, 2310, 2312, 2316, 2318, 2340, 2342, 2346, 2348, 2730, 2732, 2736, 2738, 2760, 2762, 2766, 2768] 
20220206, 22:36  #32  
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{3}×293 Posts 
Quote:
If the Primorial pattern holds then: Twin => p, p+2 1/2 => p, p+2(2#), p,4 TwinTwin => p, p+2, p+6(3#), p+8 TwinTwinTwin => p, p+2, p+6, p+8, p+30(5#), p+32, p+36, p+38 TwinTwinTwinTwin => p, p+2, p+6, p+8, p+30, p+32, p+36, p+38, p+210(7#), p+212, p+216, p+218, p+240, p+242, p+246, p+248 ... So, 11 & 19 or any other prime less than 210 might very well be missed since the constellation does not increment by 2 only. That is if I am not missing something. ETA: If I don't have the usual errors in my code, then for p = 10531061 p, p+2, p+6, p+8, p+30, p+32, p+36, p+38, p+210 are all primes: Code:
p=10531061 %1 = 10531061 (18:20) gp > isprime(p) %2 = 1 (18:21) gp > isprime(p+2) %3 = 1 (18:21) gp > isprime(p+6) %4 = 1 (18:21) gp > isprime(p+8) %5 = 1 (18:21) gp > isprime(p+30) %6 = 1 (18:21) gp > isprime(p+32) %7 = 1 (18:21) gp > isprime(p+36) %8 = 1 (18:22) gp > isprime(p+38) %9 = 1 (18:22) gp > isprime(p+210) %10 = 1 Last fiddled with by a1call on 20220206 at 23:23 

20220206, 23:41  #33  
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{3}·293 Posts 
Quote:
7503957281 9812361071 13083135641 Last fiddled with by a1call on 20220206 at 23:42 

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